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L11a429 L11a431

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 5q-5 + 10q-4 - 15q-3 + 21q-2 - 22q-1 + 24 - 19q + 14q2 - 8q3 + 4q4 - q5

A2 (sl(3)) Invariant:

q-18 - 2q-16 - q-14 + q-12 - 3q-10 + 6q-8 + 2q-6 + 4q-4 + 7q-2 + 8q2 - q4 + 3q6 + 3q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

a-2z-2 + a-2 - 2a-2z2 - 3a-2z4 - a-2z6 - 2z-2 - 3 + 4z2 + 9z4 + 5z6 + z8 + a2z-2 + 3a2 - 3a2z2 - 6a2z4 - 2a2z6 - a4 + a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 + 4a-3z3 - 10a-3z5 + 7a-3z7 + a-2z-2 - 2a-2 + a-2z2 + 2a-2z4 - 9a-2z6 + 8a-2z8 - 2a-1z-1 + a-1z + 9a-1z3 - 12a-1z5 + 6a-1z9 + 2z-2 - 5 - 3z2 + 31z4 - 39z6 + 15z8 + 2z10 - 2az-1 + 3az + 8az3 - 2az5 - 19az7 + 13az9 + a2z-2 - 6a2 - 3a2z2 + 36a2z4 - 48a2z6 + 16a2z8 + 2a2z10 + 3a3z + 7a3z3 - 11a3z5 - 7a3z7 + 7a3z9 - 2a4 - a4z2 + 12a4z4 - 21a4z6 + 9a4z8 + a5z + 3a5z3 - 10a5z5 + 5a5z7 - a6z4 + a6z6

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 = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 9 3     j = 3               10 5       j = 1             14 9         j = -1           12 14           j = -3         9 10             j = -5       6 12               j = -7     4 9                 j = -9   1 6                   j = -11   4                     j = -13 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, 430]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 430]]
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, 10, 19, 9], X[4, 17, 1, 18], X[22, 19, 13, 20], > X[10, 14, 11, 13], X[12, 21, 5, 22], X[20, 11, 21, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -7}, {2, -1, 5, -3, 6, -9, 11, -10}, > {9, -2, 4, -5, 7, -6, 8, -11, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-6 5 10 15 21 22 2 3 4 5 24 + q - -- + -- - -- + -- - -- - 19 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 -12 3 6 2 4 7 2 4 6 8 q - --- - q + q - --- + -- + -- + -- + -- + 8 q - q + 3 q + 3 q - 16 10 8 6 4 2 q q q q q q 10 12 14 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 430]][a, z]
Out[8]=	2 2 -2 2 4 2 1 a 2 2 z 2 2 4 2 4 -3 + a + 3 a - a - -- + ----- + -- + 4 z - ---- - 3 a z + a z + 9 z - 2 2 2 2 2 z a z z a 4 6 3 z 2 4 4 4 6 z 2 6 8 > ---- - 6 a z + a z + 5 z - -- - 2 a z + z 2 2 a a
In[9]:=	Kauffman[Link[11, Alternating, 430]][a, z]
Out[9]=	2 2 2 4 2 1 a 2 2 a z 3 -5 - -- - 6 a - 2 a + -- + ----- + -- - --- - --- + - + 3 a z + 3 a z + 2 2 2 2 2 a z z a a z a z z 2 2 3 3 3 5 2 2 z z 2 2 4 2 z 4 z 9 z 3 > a z - 3 z + ---- + -- - 3 a z - a z - -- + ---- + ---- + 8 a z + 4 2 5 3 a a a a a 4 4 3 3 5 3 4 6 z 2 z 2 4 4 4 6 4 > 7 a z + 3 a z + 31 z - ---- + ---- + 36 a z + 12 a z - a z + 4 2 a a 5 5 5 6 6 z 10 z 12 z 5 3 5 5 5 6 4 z 9 z > -- - ----- - ----- - 2 a z - 11 a z - 10 a z - 39 z + ---- - ---- - 5 3 a 4 2 a a a a 7 2 6 4 6 6 6 7 z 7 3 7 5 7 8 > 48 a z - 21 a z + a z + ---- - 19 a z - 7 a z + 5 a z + 15 z + 3 a 8 9 8 z 2 8 4 8 6 z 9 3 9 10 2 10 > ---- + 16 a z + 9 a z + ---- + 13 a z + 7 a z + 2 z + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 4 1 6 4 9 6 12 -- + 14 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 9 10 12 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 10 q t + 5 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a430

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