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DrawMorseLink

PD Presentation:

X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,14,19,13 X20,9,21,10 X12,19,5,20 X22,11,13,12 X10,21,11,22 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-8 + 5q-7 - 10q-6 + 16q-5 - 20q-4 + 24q-3 - 22q-2 + 20q-1 - 13 + 8q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-24 + 2q-22 + q-20 + 6q-16 - q-14 + 5q-12 + 4q-10 + 2q-8 + 8q-6 - 2q-4 + 5q-2 - 1 - 2q2 + 2q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2z2 + 3z4 + z6 + a2z-2 + 3a2 - 4a2z2 - 9a2z4 - 5a2z6 - a2z8 - 2a4z-2 - 4a4 + 3a4z2 + 6a4z4 + 2a4z6 + a6z-2 + a6 - a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 5a-1z3 - 10a-1z5 + 4a-1z7 - 6z2 + 17z4 - 19z6 + 7z8 + 6az3 - 2az5 - 9az7 + 6az9 + a2z-2 - 3a2 - 13a2z2 + 47a2z4 - 45a2z6 + 13a2z8 + 2a2z10 - 2a3z-1 + 4a3z + 3a3z3 + 7a3z5 - 22a3z7 + 13a3z9 + 2a4z-2 - 4a4 - 9a4z2 + 36a4z4 - 44a4z6 + 17a4z8 + 2a4z10 - 2a5z-1 + 4a5z + 5a5z3 - 16a5z5 + a5z7 + 7a5z9 + a6z-2 - a6 - 2a6z2 + 3a6z4 - 14a6z6 + 11a6z8 + 3a7z3 - 14a7z5 + 10a7z7 + a8 - 5a8z4 + 5a8z6 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   5 1   j = 1                 8 3     j = -1               12 5       j = -3             12 10         j = -5           12 10           j = -7         8 12             j = -9       8 12               j = -11     4 10                 j = -13   1 6                   j = -15   4                     j = -17 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, 432]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 432]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 6, 15, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[18, 14, 19, 13], X[20, 9, 21, 10], X[12, 19, 5, 20], > X[22, 11, 13, 12], X[10, 21, 11, 22], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 9, -8}, > {6, -2, 4, -5, 11, -6, 8, -7, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-8 5 10 16 20 24 22 20 2 3 -13 - q + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -20 6 -14 5 4 2 8 2 5 2 -1 - q + --- + q + --- - q + --- + --- + -- + -- - -- + -- - 2 q + 22 16 12 10 8 6 4 2 q q q q q q q q 4 6 8 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 432]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 2 4 2 6 2 4 3 a - 4 a + a + -- - ---- + -- + 2 z - 4 a z + 3 a z - a z + 3 z - 2 2 2 z z z 2 4 4 4 6 4 6 2 6 4 6 2 8 > 9 a z + 6 a z - a z + z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 432]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -3 a - 4 a - a + a + -- + ---- + -- - ---- - ---- + 4 a z + 4 a z - 2 2 2 z z z z z 3 2 2 2 4 2 6 2 5 z 3 3 3 5 3 > 6 z - 13 a z - 9 a z - 2 a z + ---- + 6 a z + 3 a z + 5 a z + a 4 5 7 3 4 2 z 2 4 4 4 6 4 8 4 10 z > 3 a z + 17 z - ---- + 47 a z + 36 a z + 3 a z - 5 a z - ----- - 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 2 a z + 7 a z - 16 a z - 14 a z + a z - 19 z + -- - 45 a z - 2 a 7 4 6 6 6 8 6 4 z 7 3 7 5 7 > 44 a z - 14 a z + 5 a z + ---- - 9 a z - 22 a z + a z + a 7 7 8 2 8 4 8 6 8 9 3 9 > 10 a z + 7 z + 13 a z + 17 a z + 11 a z + 6 a z + 13 a z + 5 9 2 10 4 10 > 7 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 12 1 4 1 6 4 10 8 12 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 8 12 12 10 12 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 5 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a432

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