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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 X22,9,13,10 X20,11,21,12 X12,19,5,20 X10,21,11,22 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-7 + 4q-6 - 6q-5 + 11q-4 - 14q-3 + 18q-2 - 17q-1 + 16 - 12q + 8q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-22 + q-20 + 3q-18 + q-16 + 6q-14 + 5q-12 + 2q-10 + 6q-8 + 2q-4 + q-2 - 1 + 4q2 - 3q4 + q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 + 1 - z2 - 2z4 - z6 + a2z-2 - 2a2z2 - 2a2z4 - a2z6 - 2a4z-2 - a4 + 3a4z2 + 2a4z4 + a6z-2 - a6z2

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 + 2a-2z2 - 8a-2z4 + 8a-2z6 + 3a-1z3 - 12a-1z5 + 10a-1z7 + 1 - 2z2 + 2z4 - 11z6 + 9z8 + 2az3 - 6az5 - 5az7 + 6az9 + a2z-2 - a2 - 14a2z2 + 39a2z4 - 38a2z6 + 9a2z8 + 2a2z10 - 2a3z-1 + 4a3z - 9a3z3 + 30a3z5 - 34a3z7 + 11a3z9 + 2a4z-2 - 4a4 - 13a4z2 + 45a4z4 - 35a4z6 + 4a4z8 + 2a4z10 - 2a5z-1 + 4a5z - 5a5z3 + 17a5z5 - 18a5z7 + 5a5z9 + a6z-2 - 3a6 - 3a6z2 + 17a6z4 - 16a6z6 + 4a6z8 + a7z3 - 3a7z5 + a7z7

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 = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 7 3     j = 1               9 5       j = -1             10 9         j = -3           8 7           j = -5         6 10             j = -7       5 8               j = -9     3 8                 j = -11   1 3                   j = -13   3                     j = -15 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, 433]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 433]]
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[22, 9, 13, 10], X[20, 11, 21, 12], > X[12, 19, 5, 20], 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, 8, -9}, > {6, -2, 4, -5, 11, -6, 9, -8, 10, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-7 4 6 11 14 18 17 2 3 4 16 - q + -- - -- + -- - -- + -- - -- - 12 q + 8 q - 4 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 3 -16 6 5 2 6 2 -2 2 4 -1 - q + q + --- + q + --- + --- + --- + -- + -- + q + 4 q - 3 q + 18 14 12 10 8 4 q q q q q q 6 8 10 12 > q + q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 433]][a, z]
Out[8]=	2 4 6 2 4 4 a 2 a a 2 z 2 2 4 2 6 2 4 z 1 - a + -- - ---- + -- - z + -- - 2 a z + 3 a z - a z - 2 z + -- - 2 2 2 2 2 z z z a a 2 4 4 4 6 2 6 > 2 a z + 2 a z - z - a z
In[9]:=	Kauffman[Link[11, Alternating, 433]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 a 2 a a 2 a 2 a 3 5 2 1 - a - 4 a - 3 a + -- + ---- + -- - ---- - ---- + 4 a z + 4 a z - 2 z + 2 2 2 z z z z z 2 3 3 2 z 2 2 4 2 6 2 2 z 3 z 3 3 3 > ---- - 14 a z - 13 a z - 3 a z - ---- + ---- + 2 a z - 9 a z - 2 3 a a a 4 4 5 3 7 3 4 z 8 z 2 4 4 4 6 4 > 5 a z + a z + 2 z + -- - ---- + 39 a z + 45 a z + 17 a z + 4 2 a a 5 5 6 4 z 12 z 5 3 5 5 5 7 5 6 8 z > ---- - ----- - 6 a z + 30 a z + 17 a z - 3 a z - 11 z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 10 z 7 3 7 5 7 > 38 a z - 35 a z - 16 a z + ----- - 5 a z - 34 a z - 18 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 5 9 > a z + 9 z + 9 a z + 4 a z + 4 a z + 6 a z + 11 a z + 5 a z + 2 10 4 10 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 3 3 8 5 8 - + 9 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 6 10 8 7 10 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 5 q t + 7 q t + 3 q t + 5 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a433

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