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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X20,14,21,13 X8,19,9,20 X18,9,19,10 X22,16,17,15 X16,18,5,17 X14,22,15,21 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-2 - 2q-1 + 5 - 7q + 11q2 - 11q3 + 13q4 - 11q5 + 9q6 - 6q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 3 + q2 + 4q4 + 3q6 + 3q8 + 6q10 + q12 + 5q14 - 2q22 + q24 - q26

HOMFLY-PT Polynomial:

- 3a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + a-4z-2 + 9a-4 + 17a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 2a-2z-2 - 10a-2 - 16a-2z2 - 10a-2z4 - 2a-2z6 + z-2 + 4 + 4z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 + a-9z - 4a-9z3 + 6a-9z5 - a-8 + 5a-8z2 - 13a-8z4 + 9a-8z6 + 6a-7z3 - 19a-7z5 + 10a-7z7 + 2a-6 - 5a-6z2 + 12a-6z4 - 22a-6z6 + 9a-6z8 - 6a-5z + 19a-5z3 - 13a-5z5 - 8a-5z7 + 5a-5z9 - a-4z-2 + 13a-4 - 44a-4z2 + 76a-4z4 - 54a-4z6 + 10a-4z8 + a-4z10 + 2a-3z-1 - 9a-3z + 3a-3z3 + 27a-3z5 - 28a-3z7 + 7a-3z9 - 2a-2z-2 + 15a-2 - 47a-2z2 + 61a-2z4 - 29a-2z6 + 2a-2z8 + a-2z10 + 2a-1z-1 - 4a-1z - 5a-1z3 + 15a-1z5 - 10a-1z7 + 2a-1z9 - z-2 + 6 - 13z2 + 13z4 - 6z6 + z8

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 5 2     j = 11               6 4       j = 9             7 5         j = 7           6 8           j = 5         5 5             j = 3       4 8               j = 1     1 3                 j = -1   1 4                   j = -3   1                     j = -5 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, 474]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 474]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[20, 14, 21, 13], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[22, 16, 17, 15], X[16, 18, 5, 17], > X[14, 22, 15, 21], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {8, -6, 5, -4, 9, -7}, > {10, -1, 3, -5, 6, -2, 11, -3, 4, -9, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 2 3 4 5 6 7 8 9 5 + q - - - 7 q + 11 q - 11 q + 13 q - 11 q + 9 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 4 6 8 10 12 14 22 3 + q + q + q + q + 4 q + 3 q + 3 q + 6 q + q + 5 q - 2 q + 24 26 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 474]][a, z]
Out[8]=	2 2 2 3 9 10 -2 1 2 2 5 z 17 z 16 z 4 4 - -- + -- - -- + z + ----- - ----- + 4 z - ---- + ----- - ----- + z - 6 4 2 4 2 2 2 6 4 2 a a a a z a z a a a 4 4 4 6 6 6 8 4 z 14 z 10 z z 6 z 2 z z > ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 474]][a, z]
Out[9]=	-8 2 13 15 -2 1 2 2 2 z 6 z 9 z 6 - a + -- + -- + -- - z - ----- - ----- + ---- + --- + -- - --- - --- - 6 4 2 4 2 2 2 3 a z 9 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 3 4 z 2 5 z 5 z 44 z 47 z z 4 z 6 z 19 z > --- - 13 z + ---- - ---- - ----- - ----- + --- - ---- + ---- + ----- + a 8 6 4 2 11 9 7 5 a a a a a a a a 3 3 4 4 4 4 4 5 5 3 z 5 z 4 3 z 13 z 12 z 76 z 61 z 6 z 19 z > ---- - ---- + 13 z + ---- - ----- + ----- + ----- + ----- + ---- - ----- - 3 a 10 8 6 4 2 9 7 a a a a a a a a 5 5 5 6 6 6 6 7 13 z 27 z 15 z 6 9 z 22 z 54 z 29 z 10 z > ----- + ----- + ----- - 6 z + ---- - ----- - ----- - ----- + ----- - 5 3 a 8 6 4 2 7 a a a a a a a 7 7 7 8 8 8 9 9 9 8 z 28 z 10 z 8 9 z 10 z 2 z 5 z 7 z 2 z > ---- - ----- - ----- + z + ---- + ----- + ---- + ---- + ---- + ---- + 5 3 a 6 4 2 5 3 a a a a a a a a 10 10 z z > --- + --- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 4 q 3 q 4 q 5 7 8 q + 5 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 5 q t + 6 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 8 q t + 7 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 15 5 15 6 17 6 19 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a474

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