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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 7q-1 + 11 - 11q + 12q2 - 10q3 + 8q4 - 5q5 + 3q6 - q7

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + 2q-6 + 6q-2 + 4 + 4q2 + 4q4 + q6 + 3q8 - q10 + 2q14 - 2q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4z4 + a-2z-2 + 2a-2 + 2a-2z2 + 2a-2z4 - 2z-2 - 3 - 2z2 + z4 + a2z-2 - 2a2z2 + a4

Kauffman Polynomial:

4a-7z3 - 4a-7z5 + a-7z7 - 5a-6z2 + 16a-6z4 - 13a-6z6 + 3a-6z8 - 4a-5z3 + 11a-5z5 - 11a-5z7 + 3a-5z9 - 3a-4z2 + 14a-4z4 - 15a-4z6 + 2a-4z8 + a-4z10 - 4a-3z3 + 10a-3z5 - 14a-3z7 + 5a-3z9 + a-2z-2 - 5a-2 + 9a-2z2 - 4a-2z4 - 6a-2z6 + 2a-2z8 + a-2z10 - 2a-1z-1 + 6a-1z + a-1z3 - 7a-1z5 + a-1z7 + 2a-1z9 + 2z-2 - 8 + 12z2 - 6z4 - z6 + 3z8 - 2az-1 + 6az - 5az3 + 3az7 + a2z-2 - 3a2 + 3a2z2 - 3a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

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 = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 5 2     j = 7               5 3       j = 5             7 5         j = 3           6 7           j = 1         5 5             j = -1       3 7               j = -3     2 4                 j = -5     3                   j = -7 1 2                     j = -9 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, 418]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 418]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 10, 19, 9], X[16, 8, 17, 7], > X[22, 14, 11, 13], X[20, 16, 21, 15], X[10, 18, 5, 17], X[8, 20, 9, 19], > X[14, 22, 15, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -8, 3, -7}, > {11, -2, 5, -9, 6, -4, 7, -3, 8, -6, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 5 7 2 3 4 5 6 7 11 + q - -- + -- - - - 11 q + 12 q - 10 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 2 6 2 4 6 8 10 14 4 + q + q - q + -- + -- + -- + 4 q + 4 q + q + 3 q - q + 2 q - 8 6 2 q q q 16 18 20 22 > 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 418]][a, z]
Out[8]=	2 2 2 4 4 2 4 2 1 a 2 z 2 z 2 2 4 z 2 z -3 + -- + a - -- + ----- + -- - 2 z - -- + ---- - 2 a z + z + -- + ---- 2 2 2 2 2 6 2 4 2 a z a z z a a a a
In[9]:=	Kauffman[Link[11, Alternating, 418]][a, z]
Out[9]=	2 5 2 4 2 1 a 2 2 a 6 z 2 -8 - -- - 3 a + a + -- + ----- + -- - --- - --- + --- + 6 a z + 12 z - 2 2 2 2 2 a z z a a z a z z 2 2 2 3 3 3 3 5 z 3 z 9 z 2 2 4 2 4 z 4 z 4 z z 3 > ---- - ---- + ---- + 3 a z - 2 a z + ---- - ---- - ---- + -- - 5 a z - 6 4 2 7 5 3 a a a a a a a 4 4 4 5 5 3 3 4 16 z 14 z 4 z 2 4 4 4 4 z 11 z > 2 a z - 6 z + ----- + ----- - ---- - 3 a z + a z - ---- + ----- + 6 4 2 7 5 a a a a a 5 5 6 6 6 7 7 10 z 7 z 3 5 6 13 z 15 z 6 z 2 6 z 11 z > ----- - ---- + 2 a z - z - ----- - ----- - ---- + 3 a z + -- - ----- - 3 a 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 9 14 z z 7 8 3 z 2 z 2 z 3 z 5 z 2 z > ----- + -- + 3 a z + 3 z + ---- + ---- + ---- + ---- + ---- + ---- + 3 a 6 4 2 5 3 a a a a a a a 10 10 z z > --- + --- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 1 2 3 2 4 3 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 5 q t + 6 q t + q 9 4 7 4 7 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 7 q t + 7 q t + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a418

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