# Simplifying the following mathematical expression using a computer?

I have this following beastly expression typed up very nicely in LaTeX formatting, as you can see. What is the easiest way that I can get a computer to simplify this expression for me? I have zero programming experience. I installed sagemath but it seems pretty complicated.

$$W_{(1,1)}(t,v)=\frac{-t^{-2k}v^k}{3}(\frac{v^{\frac{3}{2}}-v^{\frac{-3}{2}}}{t^{\frac{3}{2}}-t^{\frac{-3}{2}}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})+\frac{t^{-2k}v^k}{4}(\frac{v-v^{-1}}{t-t^{-1}})^2+\frac{t^{-2k}v^k}{12}(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})-\frac{t^{-k}v^k}{4}(\frac{v^2-v^{-2}}{t^2-t^{-2}}) + \frac{t^{-k}v^k}{8}(\frac{v-v^{-1}}{t-t^{-1}})^2+\frac{t^{-k}v^k}{4}(\frac{v-v^{-1}}{t-t^{-1}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^2-\frac{t^{-k}v^k}{8}(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^4+\frac{-v^kt^{k}}{4}(\frac{v^2-v^{-2}}{t^2-t^{-2}})+\frac{v^kt^{k}}{3}(\frac{v^{\frac{3}{2}}-v^{\frac{-3}{2}}}{t^{\frac{3}{2}}-t^{\frac{-3}{2}}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})+\frac{v^kt^{k}}{8}(\frac{v-v^{-1}}{t-t^{-1}})^2-\frac{v^kt^{k}}{4}(\frac{v-v^{-1}}{t-t^{-1}})(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^2+\frac{v^kt^{k}}{24}(\frac{v^{\frac{1}{2}}-v^{\frac{-1}{2}}}{t^{\frac{1}{2}}-t^{\frac{-1}{2}}})^4$$

• what is this an expression of Sep 17, 2020 at 0:45
• Have you tried Mathematica ? Sep 17, 2020 at 1:01
• Mathematica is certainly not free, in any sense of the word. Sep 17, 2020 at 2:15
• I've tried this. Mathematica can, say, make a single fraction out of this, but the numerator involves a polynomial jumble with scores of terms, which is by no means "simple". Maybe one can combine pieces of the expression to reduce the overall complexity, but that kind of thing requires human judgment and finesse. Before I would invest any time in such a problem, I'd have to believe there's a worthwhile outcome. Do other "colored homfly polynomials" have a compact form? Also, have you checked the expression for typos? (Can you describe how it was generated, so that we can check your work?)
– Blue
Sep 17, 2020 at 16:14
• I might also suggest introducing notation like, say, $$X_{p}:=\frac{v^p-v^{-p}}{t^p-t^{-p}}$$ This would greatly reduce visual clutter (and the possibility of typographic and transcription errors) by allowing one to write $$\frac{-t^{-2k}v^k}{3}X_{3/2}X_{1/2}+\frac{t^{-2k}v^k}{4}X_1^2+\frac{t^{-2k}v^k}{12}X_{1/2}-\frac{t^{-k}v^k}{4}X_2 +\cdots$$
– Blue
Sep 17, 2020 at 16:39

Here's an all-form-and-no-substance simplification that merely introduces the notation $$X_p:= \frac{v^p-v^{-p}}{t^p-t^{-p}}$$ and factors-out powers of $$t$$ and $$v$$ from certain groups:

\begin{align} W_{(1,1)}(t,v)&=\frac{t^{-2k}v^k}{12}\left( -4X_{3/2}X_{1/2}+3X_1^2+X_{1/2}^4\right) \\[4pt] &+\frac{t^{-k} v^k}{8}\left(-2X_2 + X_1^2+2X_1X_{1/2}^2-X_{1/2}^4\right) \\[4pt] &+\frac{t^k v^k}{24}\left(-6X_2 +8X_{3/2}X_{1/2}+3X_1^2-6X_1X_{1/2}^2+X_{1/2}^4\right) \end{align} \tag{1}

That the $$X$$-subscripts in each term (interpreting exponents as multipliers) sum to $$2$$ serves as a nice sanity-check. (OP's original rendering of the expression omitted the exponent on the $$X_{1/2}$$ term in the first line. The subscript check helped identify this error. Check the edit history for a version of this answer that included consideration of the erroneous expression.)

Dropping $$(1)$$ into Mathematica makes a bit of a mess. To de-messify it slightly, define $$s := \sqrt{t} \qquad u := \sqrt{v}$$ so that half-powers of $$t$$ and $$v$$ become integer powers of $$s$$ and $$u$$. Then, simplifying $$(1)$$ grouping-by-grouping gives \begin{align} W_{(1,1)}(s^2,u^2) = &-s^{-4k+6}u^{2k-4} \frac{(u^2-1)^2 (s^2-u^2) (s^2 u^2-1)}{(s^2-1) (s^4-1)^2 (s^6-1)} \\[4pt] &-s^{-2k+4}u^{2k-4}\frac{\left(u^2-1\right) \left(s^2-u^2\right) \left(s^2 u^2-1\right)(s^4-u^2)}{(s^2-1)^2(s^4-1) (s^8-1)} \\[4pt] &-s^{2k+4} u^{2k-4}\frac{(u^2-1)(s^2-u^2)(s^4-u^2)(s^6-u^2)}{(s^2-1) (s^4-1) (s^6-1) (s^8-1)} \\[4pt] =&\phantom{-}\frac{u^{2k}}{s^{4k}}\;\frac{u_1^2 m_1 m_{-1}}{s_1 s_2^2 s_3} +\frac{u^{2k}}{s^{2k}}\;\frac{u_1 m_1 m_2m_{-1} }{s_1^2 s_2 s_4} -s^{2k}u^{2k}\;\frac{u_1 m_1 m_2 m_3}{s_1 s_2 s_3 s_4} \\[4pt] =&\phantom{-}\frac{u^{2k}}{s^{4k}}\;\frac{ u_1m_1(s_1 s_4 u_1 m_{-1} +s^{2k}s_2 s_3 m_2 m_{-1} -s^{6k}s_1 s_2 m_2 m_3) }{s_1^2s_2^2s_3 s_4} \end{align} where $$s_p:=s^p-s^{-p} \qquad u_p:=u^p-u^{-p} \qquad m_p := \frac{s^p}{u}-\frac{u}{s^p}$$

Further defining $$r_p := s^p+s^{-p}$$, we have \begin{align} s_3 &= s^3-s^{-3}= (s-s^{-1})(s^2+1+s^{-2}) = s_1 (r_2+1) \\ s_4 &= s^4-s^{-4}= (s^2-s^{-2})(s^2+s^{-2}) = s_2 r_2 \end{align} which allows us to write

$$W_{(1,1)}(s^2,u^2)=\frac{u^{2k}}{s^{4k}}\;\frac{ u_1m_1(r_2 u_1 m_{-1} +s^{2k} (r_2+1) m_2 m_{-1} -s^{6k} m_2 m_3) }{s_1 s_2 s_3 s_4}$$

There may be even cleaner ways to write the expression, but this is as far as I'll go.

• Yes, it should have been to the fourth, good catch. You are a complete boss. I've gotten MatLab up and running and i'm digging it. Thanks for your insights man!!
– user637978
Sep 17, 2020 at 23:29

Wolfram|Alpha supports Latex. Unfortunately, the engine didn't seem to understand your query, perhaps because of its length. If possible, I would suggest splitting the expression up into chunks, and entering them one by one into Wolfram|Alpha.