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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 9 + 15q - 17q2 + 19q3 - 16q4 + 13q5 - 9q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 5q-4 + 2q-2 + 3 + 7q2 + 5q6 + q8 + q10 + 3q12 - 3q14 + 3q16 - 2q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + a-4 + 2a-4z2 + 2a-4z4 + a-4z6 + a-2z-2 + 3a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 2z-2 - 5 - 5z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 - 5a-8z4 + 4a-8z6 - 2a-7z + 6a-7z3 - 14a-7z5 + 8a-7z7 + 2a-6 - 7a-6z2 + 15a-6z4 - 18a-6z6 + 9a-6z8 - 3a-5z + 8a-5z3 - 7a-5z5 - 2a-5z7 + 5a-5z9 + 2a-4 - 13a-4z2 + 31a-4z4 - 31a-4z6 + 12a-4z8 + a-4z10 + 3a-3z - 5a-3z3 + 8a-3z5 - 11a-3z7 + 7a-3z9 + a-2z-2 - 6a-2 + 7a-2z2 - 9a-2z6 + 5a-2z8 + a-2z10 - 2a-1z-1 + 7a-1z - 4a-1z3 - 5a-1z5 + a-1z7 + 2a-1z9 + 2z-2 - 9 + 19z2 - 15z4 + z6 + 2z8 - 2az-1 + 3az + 2az3 - 5az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + 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                     3   j = 13                   6 1   j = 11                 7 3     j = 9               9 6       j = 7             10 7         j = 5           7 9           j = 3         8 10             j = 1       5 11               j = -1     1 4                 j = -3   1 5                   j = -5   1                     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, 465]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 465]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[20, 13, 21, 14], > X[16, 10, 17, 9], X[8, 16, 9, 15], X[22, 18, 15, 17], X[18, 22, 19, 21], > X[14, 19, 5, 20], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, 7, -8, 9, -4, 8, -7}, > {10, -1, 3, -6, 5, -2, 11, -3, 4, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 6 2 3 4 5 6 7 8 -9 + q - -- + - + 15 q - 17 q + 19 q - 16 q + 13 q - 9 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 5 2 2 6 8 10 12 14 3 + q + q + q + -- + -- + 7 q + 5 q + q + q + 3 q - 3 q + 4 2 q q 16 18 20 22 24 > 3 q - 2 q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 465]][a, z]
Out[8]=	2 2 2 2 -6 -4 3 2 2 1 a 2 z 2 z 3 z -5 - a + a + -- + 2 a - -- + ----- + -- - 5 z - -- + ---- + ---- + 2 2 2 2 2 6 4 2 a z a z z a a a 4 4 4 6 6 2 2 4 z 2 z 2 z z z > a z - 2 z - -- + ---- + ---- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 465]][a, z]
Out[9]=	2 2 2 6 2 2 1 a 2 2 a 2 z 3 z 3 z -9 + -- + -- - -- - 4 a + -- + ----- + -- - --- - --- - --- - --- + --- + 6 4 2 2 2 2 2 a z z 7 5 3 a a a z a z z a a a 2 2 2 3 3 3 7 z 2 7 z 13 z 7 z 2 2 z 6 z 8 z > --- + 3 a z + 19 z - ---- - ----- + ---- + 6 a z - -- + ---- + ---- - a 6 4 2 9 7 5 a a a a a a 3 3 4 4 4 5 5 z 4 z 3 4 5 z 15 z 31 z 2 4 z > ---- - ---- + 2 a z - 15 z - ---- + ----- + ----- - 4 a z + -- - 3 a 8 6 4 9 a a a a a 5 5 5 5 6 6 6 6 14 z 7 z 8 z 5 z 5 6 4 z 18 z 31 z 9 z > ----- - ---- + ---- - ---- - 5 a z + z + ---- - ----- - ----- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 2 6 8 z 2 z 11 z z 7 8 9 z 12 z 5 z > a z + ---- - ---- - ----- + -- + 2 a z + 2 z + ---- + ----- + ---- + 7 5 3 a 6 4 2 a a a a a a 9 9 9 10 10 5 z 7 z 2 z z z > ---- + ---- + ---- + --- + --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 5 1 4 5 q 3 11 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q t + 7 4 5 3 3 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 > 7 q t + 9 q t + 10 q t + 7 q t + 9 q t + 6 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a465

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