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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X20,5,21,6 X18,9,19,10 X22,19,9,20 X16,12,17,11 X6,21,7,22 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 4q-9/2 - 8q-7/2 + 12q-5/2 - 16q-3/2 + 17q-1/2 - 18q1/2 + 15q3/2 - 11q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-16 - 2q-14 + q-12 - q-8 + 4q-6 - 2q-4 + 3q-2 + 1 + 4q4 - 2q6 + 3q8 - q12 + q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 3a-3z3 + a-3z5 - a-1z-1 - 4a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + 2az - az3 - 3az5 - az7 + 2a3z3 + a3z5

Kauffman Polynomial:

a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 - a-4z2 + 4a-4z4 - 5a-4z6 + a-3z - 4a-3z3 + 8a-3z5 - 7a-3z7 + a-2z2 - 8a-2z4 + 13a-2z6 - 8a-2z8 - a-1z-1 + 6a-1z - 15a-1z3 + 10a-1z5 + 6a-1z7 - 6a-1z9 + 1 + 9z2 - 37z4 + 41z6 - 10z8 - 2z10 - az-1 + 6az - 11az3 - 9az5 + 28az7 - 11az9 + 10a2z2 - 38a2z4 + 37a2z6 - 6a2z8 - 2a2z10 + 3a3z - 6a3z3 - 5a3z5 + 14a3z7 - 5a3z9 + 4a4z2 - 14a4z4 + 14a4z6 - 4a4z8 + a5z - 3a5z3 + 3a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     2   j = 8                   4 1   j = 6                 7 2     j = 4               8 4       j = 2             10 7         j = 0           9 10           j = -2         7 8             j = -4       5 9               j = -6     3 7                 j = -8   1 5                   j = -10   3                     j = -12 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 328]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 328]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 5, 21, 6], X[18, 9, 19, 10], > X[22, 19, 9, 20], X[16, 12, 17, 11], X[6, 21, 7, 22], X[14, 8, 15, 7], > X[4, 14, 5, 13], X[8, 16, 1, 15], X[2, 17, 3, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 3, -7, 8, -10}, > {4, -1, 6, -2, 9, -8, 10, -6, 11, -4, 5, -3, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 12 16 17 3/2 -q + ---- - ---- + ---- - ---- + ------- - 18 Sqrt[q] + 15 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 11 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 -8 4 2 3 4 6 8 12 14 1 + q - --- + q - q + -- - -- + -- + 4 q - 2 q + 3 q - q + q - 14 6 4 2 q q q q 16 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 328]][a, z]
Out[8]=	3 3 5 5 1 a 2 z 4 z 3 z 6 z 3 3 3 z 4 z -(---) + - + --- - --- + 2 a z + ---- - ---- - a z + 2 a z + -- - ---- - a z z 3 a 3 a 3 a a a a 7 5 3 5 z 7 > 3 a z + a z - -- - a z a
In[9]:=	Kauffman[Link[11, Alternating, 328]][a, z]
Out[9]=	2 2 2 1 a z z 6 z 3 5 2 z z z 1 - --- - - - -- + -- + --- + 6 a z + 3 a z + a z + 9 z + -- - -- + -- + a z z 5 3 a 6 4 2 a a a a a 3 3 3 2 2 4 2 3 z 4 z 15 z 3 3 3 5 3 > 10 a z + 4 a z + ---- - ---- - ----- - 11 a z - 6 a z - 3 a z - 5 3 a a a 4 4 4 5 5 5 4 z 4 z 8 z 2 4 4 4 3 z 8 z 10 z > 37 z - -- + ---- - ---- - 38 a z - 14 a z - ---- + ---- + ----- - 6 4 2 5 3 a a a a a a 6 6 5 3 5 5 5 6 5 z 13 z 2 6 4 6 > 9 a z - 5 a z + 3 a z + 41 z - ---- + ----- + 37 a z + 14 a z - 4 2 a a 7 7 8 7 z 6 z 7 3 7 5 7 8 8 z 2 8 > ---- + ---- + 28 a z + 14 a z - a z - 10 z - ---- - 6 a z - 3 a 2 a a 9 4 8 6 z 9 3 9 10 2 10 > 4 a z - ---- - 11 a z - 5 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 7 5 9 10 + 10 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 7 9 8 2 4 4 2 6 2 6 3 > ----- + - + ---- + 7 q t + 8 q t + 4 q t + 7 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a328

L11a327 L11a329