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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X16,6,17,5 X8493 X20,9,21,10 X10,19,11,20 X22,18,15,17 X18,22,19,21 X14,16,5,15 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 12q-2 - 16q-1 + 21 - 20q + 19q2 - 14q3 + 10q4 - 5q5 + q6

A2 (sl(3)) Invariant:

- q-16 + q-12 - 2q-10 + 4q-8 + 4q-6 + 7q-2 + 1 + 5q2 + 4q4 + q6 + 6q8 - 3q10 + q12 + q14 - 3q16 + q18

HOMFLY-PT Polynomial:

- a-4 + a-4z4 + a-2z-2 + 4a-2 + 2a-2z2 - a-2z4 - a-2z6 - 2z-2 - 6 - 5z2 - 2z4 - z6 + a2z-2 + 4a2 + 4a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

- a-6z4 + a-6z6 + a-5z + 2a-5z3 - 10a-5z5 + 5a-5z7 - 2a-4 + a-4z2 + 11a-4z4 - 22a-4z6 + 9a-4z8 + 3a-3z + a-3z3 - 2a-3z5 - 11a-3z7 + 7a-3z9 + a-2z-2 - 8a-2 + 10a-2z2 + 16a-2z4 - 33a-2z6 + 11a-2z8 + 2a-2z10 - 2a-1z-1 + 4a-1z - a-1z3 + 9a-1z5 - 20a-1z7 + 11a-1z9 + 2z-2 - 11 + 22z2 - 7z4 - 10z6 + 6z8 + 2z10 - 2az-1 + 4az - az3 - 3az5 + 4az9 + a2z-2 - 8a2 + 18a2z2 - 17a2z4 + 3a2z6 + 4a2z8 + 3a3z - 3a3z3 - 3a3z5 + 4a3z7 - 2a4 + 5a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     4   j = 9                   6 1   j = 7                 8 4     j = 5               11 6       j = 3             9 8         j = 1           12 11           j = -1         8 13             j = -3       4 8               j = -5     2 8                 j = -7   1 4                   j = -9   2                     j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 464]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 464]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[16, 6, 17, 5], > X[8, 4, 9, 3], X[20, 9, 21, 10], X[10, 19, 11, 20], X[22, 18, 15, 17], > X[18, 22, 19, 21], X[14, 16, 5, 15], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {10, -4, 8, -9, 7, -6, 9, -8}, > {4, -1, 2, -5, 6, -7, 11, -2, 3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 6 12 16 2 3 4 5 6 21 - q + -- - -- + -- - -- - 20 q + 19 q - 14 q + 10 q - 5 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -12 2 4 4 7 2 4 6 8 10 12 1 - q + q - --- + -- + -- + -- + 5 q + 4 q + q + 6 q - 3 q + q + 10 8 6 2 q q q q 14 16 18 > q - 3 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 464]][a, z]
Out[8]=	2 2 -4 4 2 4 2 1 a 2 2 z 2 2 4 2 -6 - a + -- + 4 a - a - -- + ----- + -- - 5 z + ---- + 4 a z - a z - 2 2 2 2 2 2 a z a z z a 4 4 6 4 z z 2 4 6 z > 2 z + -- - -- + 2 a z - z - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 464]][a, z]
Out[9]=	2 2 8 2 4 2 1 a 2 2 a z 3 z 4 z -11 - -- - -- - 8 a - 2 a + -- + ----- + -- - --- - --- + -- + --- + --- + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 3 3 5 2 z 10 z 2 2 4 2 2 z > 4 a z + 3 a z + a z + 22 z + -- + ----- + 18 a z + 5 a z + ---- + 4 2 5 a a a 3 3 4 4 4 z z 3 3 3 5 3 4 z 11 z 16 z 2 4 > -- - -- - a z - 3 a z - 2 a z - 7 z - -- + ----- + ----- - 17 a z - 3 a 6 4 2 a a a a 5 5 5 6 4 4 10 z 2 z 9 z 5 3 5 5 5 6 z > 6 a z - ----- - ---- + ---- - 3 a z - 3 a z + a z - 10 z + -- - 5 3 a 6 a a a 6 6 7 7 7 22 z 33 z 2 6 4 6 5 z 11 z 20 z 3 7 8 > ----- - ----- + 3 a z + 3 a z + ---- - ----- - ----- + 4 a z + 6 z + 4 2 5 3 a a a a a 8 8 9 9 10 9 z 11 z 2 8 7 z 11 z 9 10 2 z > ---- + ----- + 4 a z + ---- + ----- + 4 a z + 2 z + ----- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 2 1 4 2 8 4 8 -- + 12 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 11 q t + 9 q t + 8 q t + 11 q t + 6 q t + 8 q t + 4 q t + q t 9 4 9 5 11 5 13 6 > 6 q t + q t + 4 q t + q t

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