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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 9q-1 - 13 + 19q - 20q2 + 21q3 - 18q4 + 13q5 - 7q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-10 - q-6 + 5q-4 + 4 + 9q2 + 2q4 + 9q6 + q8 + 2q10 - 5q14 + 3q16 - 2q18 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 - a-6z-2 + 2a-6 + 3a-6z2 + 4a-4z-2 + 4a-4 - 2a-4z2 - 3a-4z4 - 5a-2z-2 - 9a-2 - 5a-2z2 + a-2z6 + 2z-2 + 3 - z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 3a-8z6 + a-7z-1 - 2a-7z + 3a-7z3 - 6a-7z5 + 5a-7z7 - a-6z-2 + 4a-6z2 - a-6z4 - 5a-6z6 + 6a-6z8 + 5a-5z-1 - 21a-5z + 37a-5z3 - 29a-5z5 + 7a-5z7 + 4a-5z9 - 4a-4z-2 + 14a-4 - 22a-4z2 + 35a-4z4 - 38a-4z6 + 16a-4z8 + a-4z10 + 9a-3z-1 - 41a-3z + 68a-3z3 - 48a-3z5 + 3a-3z7 + 8a-3z9 - 5a-2z-2 + 23a-2 - 44a-2z2 + 52a-2z4 - 46a-2z6 + 16a-2z8 + a-2z10 + 5a-1z-1 - 23a-1z + 39a-1z3 - 32a-1z5 + 4a-1z7 + 4a-1z9 - 2z-2 + 10 - 18z2 + 18z4 - 15z6 + 6z8 + 3az3 - 6az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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 = 17                       1 j = 15                     2   j = 13                   5 1   j = 11                 8 2     j = 9               11 6       j = 7             10 7         j = 5           10 11           j = 3         9 10             j = 1       6 12               j = -1     3 7                 j = -3     6                   j = -5 1 3                     j = -7 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, 435]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 435]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 19, 11], X[14, 8, 15, 7], > X[8, 14, 9, 13], X[22, 20, 13, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[12, 18, 5, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, > {5, -4, 7, -8, 9, -3, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 9 2 3 4 5 6 7 8 -13 + q - -- + - + 19 q - 20 q + 21 q - 18 q + 13 q - 7 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 5 2 4 6 8 10 14 16 18 4 + q - q + -- + 9 q + 2 q + 9 q + q + 2 q - 5 q + 3 q - 2 q - 4 q 20 22 24 26 > q + 2 q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 435]][a, z]
Out[8]=	2 2 -8 2 4 9 2 2 1 4 5 2 3 z 2 z 3 - a + -- + -- - -- + a + -- - ----- + ----- - ----- - z + ---- - ---- - 6 4 2 2 6 2 4 2 2 2 6 4 a a a z a z a z a z a a 2 4 6 5 z 2 2 4 3 z z > ---- + a z - 2 z - ---- + -- 2 4 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 435]][a, z]
Out[9]=	-8 14 23 2 2 1 4 5 1 5 9 10 - a + -- + -- - a - -- - ----- - ----- - ----- + ---- + ---- + ---- + 4 2 2 6 2 4 2 2 2 7 5 3 a a z a z a z a z a z a z a z 2 2 2 2 5 z 2 z 21 z 41 z 23 z 2 3 z 4 z 22 z 44 z > --- + -- - --- - ---- - ---- - ---- - 18 z + ---- + ---- - ----- - ----- + a z 9 7 5 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 3 4 2 2 2 z 3 z 37 z 68 z 39 z 3 4 5 z > 3 a z - ---- + ---- + ----- + ----- + ----- + 3 a z + 18 z - ---- - 9 7 5 3 a 8 a a a a a 4 4 4 5 5 5 5 5 z 35 z 52 z 2 4 z 6 z 29 z 48 z 32 z 5 > -- + ----- + ----- - 3 a z + -- - ---- - ----- - ----- - ----- - 6 a z - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 6 3 z 5 z 38 z 46 z 2 6 5 z 7 z 3 z 4 z > 15 z + ---- - ---- - ----- - ----- + a z + ---- + ---- + ---- + ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 6 z 16 z 16 z 4 z 8 z 4 z z z > 3 a z + 6 z + ---- + ----- + ----- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 3 6 3 7 6 q 3 12 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 10 q t + 11 q t + 10 q t + 7 q t + 11 q t + 6 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a435

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