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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 6q-9/2 + 10q-7/2 - 15q-5/2 + 17q-3/2 - 17q-1/2 + 16q1/2 - 12q3/2 + 7q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 4q-14 - 3q-12 + q-10 + q-8 - 3q-6 + 2q-4 - 3q-2 + 2 - q4 + 5q6 - 2q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - a-1z-1 - 3a-1z - a-1z3 - a-1z5 + 2az-1 + 4az + 2az3 - az5 - a3z-1 + 3a3z3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

2a-4z4 - a-4z6 - 7a-3z3 + 11a-3z5 - 4a-3z7 + 4a-2z2 - 12a-2z4 + 16a-2z6 - 6a-2z8 - a-1z-1 + 6a-1z - 18a-1z3 + 16a-1z5 + 2a-1z7 - 4a-1z9 - 1 + 15z2 - 35z4 + 35z6 - 11z8 - z10 - 2az-1 + 13az - 23az3 + 10az5 + 8az7 - 7az9 - 3a2 + 16a2z2 - 30a2z4 + 25a2z6 - 9a2z8 - a2z10 - a3z-1 + 7a3z - 12a3z3 + 8a3z5 - a3z7 - 3a3z9 - 2a4 + 5a4z2 - 6a4z4 + 5a4z6 - 4a4z8 + a5z-1 - 3a5z + 3a5z3 + 2a5z5 - 3a5z7 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

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 = 10                       1 j = 8                     3   j = 6                   4 1   j = 4                 8 3     j = 2               8 4       j = 0             9 8         j = -2           9 9           j = -4         6 8             j = -6       4 9               j = -8     2 6                 j = -10   1 5                   j = -12   1                     j = -14 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, 119]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 119]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 11, 21, 12], X[10, 21, 11, 22], X[16, 10, 17, 9], X[12, 16, 13, 15], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 7, -6, 5, -8, 11, -2, 8, -7, 9, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 6 10 15 17 17 -q + ----- - ---- + ---- - ---- + ---- - ------- + 16 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 12 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 4 3 -10 -8 3 2 3 4 6 2 + q + --- + --- + --- - --- + q + q - -- + -- - -- - q + 5 q - 20 16 14 12 6 4 2 q q q q q q q 8 12 14 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 119]][a, z]
Out[8]=	3 5 7 3 3 1 2 a a a a 3 z 5 z z 3 -(---) + --- - -- - -- + -- - --- + 4 a z - 3 a z + -- - -- + 2 a z + a z z z z z a 3 a a 5 3 3 z 5 > 3 a z - -- - a z a
In[9]:=	Kauffman[Link[11, Alternating, 119]][a, z]
Out[9]=	3 5 7 2 4 6 1 2 a a a a 6 z 3 -1 - 3 a - 2 a - a - --- - --- - -- + -- + -- + --- + 13 a z + 7 a z - a z z z z z a 2 3 3 5 7 2 4 z 2 2 4 2 7 z 18 z > 3 a z - 3 a z + 15 z + ---- + 16 a z + 5 a z - ---- - ----- - 2 3 a a a 4 4 3 3 3 5 3 7 3 4 2 z 12 z 2 4 > 23 a z - 12 a z + 3 a z + 3 a z - 35 z + ---- - ----- - 30 a z - 4 2 a a 5 5 4 4 6 4 11 z 16 z 5 3 5 5 5 7 5 > 6 a z + 3 a z + ----- + ----- + 10 a z + 8 a z + 2 a z - a z + 3 a a 6 6 7 7 6 z 16 z 2 6 4 6 6 6 4 z 2 z 7 > 35 z - -- + ----- + 25 a z + 5 a z - 2 a z - ---- + ---- + 8 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 6 z 2 8 4 8 4 z 9 > a z - 3 a z - 11 z - ---- - 9 a z - 4 a z - ---- - 7 a z - 2 a a 3 9 10 2 10 > 3 a z - z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 1 1 5 2 6 4 9 9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 6 8 9 2 2 2 4 2 4 3 > ----- + ---- + ---- + 8 t + 8 q t + 4 q t + 8 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a119

L11a118 L11a120