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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X14,21,15,22 X16,5,17,6 X18,8,19,7 X20,17,21,18 X4,15,5,16 X6,20,7,19 X22,13,9,14 X2,9,3,10 X8,11,1,12

Gauss Code:

{{1, -10, 2, -7, 4, -8, 5, -11}, {10, -1, 11, -2, 9, -3, 7, -4, 6, -5, 8, -6, 3, -9}}

Jones Polynomial:

q-21/2 - 3q-19/2 + 6q-17/2 - 10q-15/2 + 13q-13/2 - 16q-11/2 + 16q-9/2 - 14q-7/2 + 10q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-32 + q-30 - 2q-26 + 3q-24 - q-22 + 3q-18 - q-16 + 3q-14 - q-12 + q-10 + 3q-8 - 2q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - 3a3z + a3z5 + a5z-1 + 3a5z + 4a5z3 + 2a5z5 + a7z3 + a7z5 - a9z - a9z3

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 - a3z-1 + 5a3z - 6a3z3 + 9a3z5 - 5a3z7 + a4 - 3a4z4 + 7a4z6 - 5a4z8 - a5z-1 + 8a5z - 18a5z3 + 14a5z5 - 2a5z7 - 3a5z9 + 2a6z2 - 16a6z4 + 18a6z6 - 7a6z8 - a6z10 + 3a7z - 10a7z3 + a7z5 + 9a7z7 - 6a7z9 + 5a8z2 - 17a8z4 + 19a8z6 - 6a8z8 - a8z10 + 3a9z - 7a9z3 + 6a9z5 + 3a9z7 - 3a9z9 + 2a10z2 - 6a10z4 + 10a10z6 - 4a10z8 + 2a11z - 7a11z3 + 9a11z5 - 3a11z7 - 2a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     2   j = -2                   5 1   j = -4                 6 3     j = -6               8 4       j = -8             8 6         j = -10           8 8           j = -12         6 9             j = -14       4 7               j = -16     2 6                 j = -18   1 4                   j = -20   2                     j = -22 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 297]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 297]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 21, 15, 22], X[16, 5, 17, 6], > X[18, 8, 19, 7], X[20, 17, 21, 18], X[4, 15, 5, 16], X[6, 20, 7, 19], > X[22, 13, 9, 14], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -7, 4, -8, 5, -11}, > {10, -1, 11, -2, 9, -3, 7, -4, 6, -5, 8, -6, 3, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 10 13 16 16 14 10 7 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 3 -22 3 -16 3 -12 -10 3 -1 - q + q - --- + --- - q + --- - q + --- - q + q + -- - 26 24 18 14 8 q q q q q 2 3 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 297]][a, z]
Out[8]=	3 5 a a 3 5 9 3 5 3 7 3 9 3 -(--) + -- - a z - 3 a z + 3 a z - a z - a z + 4 a z + a z - a z + z z 3 5 5 5 7 5 > a z + 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 297]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 11 2 2 a - -- - -- - a z + 5 a z + 8 a z + 3 a z + 3 a z + 2 a z - a z + z z 6 2 8 2 10 2 12 2 3 3 3 5 3 > 2 a z + 5 a z + 2 a z - 2 a z + 2 a z - 6 a z - 18 a z - 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 10 a z - 7 a z - 7 a z + 5 a z - 3 a z - 16 a z - 17 a z - 10 4 12 4 5 3 5 5 5 7 5 9 5 > 6 a z + 3 a z - a z + 9 a z + 14 a z + a z + 6 a z + 11 5 2 6 4 6 6 6 8 6 10 6 12 6 > 9 a z - 3 a z + 7 a z + 18 a z + 19 a z + 10 a z - a z - 3 7 5 7 7 7 9 7 11 7 4 8 6 8 > 5 a z - 2 a z + 9 a z + 3 a z - 3 a z - 5 a z - 7 a z - 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 6 a z - 4 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 4 2 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 7 6 9 8 8 8 6 8 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 6 t 2 2 > ---- + 2 t + -- + q t 4 2 q t q

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