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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 16q-2 - 20q-1 + 24 - 22q + 21q2 - 14q3 + 8q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 4q-10 + 3q-8 + 2q-6 + 9q-2 + 2 + 8q2 + 3q4 + q6 + 5q8 - 5q10 + q12 + q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 + a-2z-2 - 3a-2z2 - 2a-2z4 - a-2z6 - 2z-2 + 2z2 - z6 + a2z-2 + a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

- 2a-6z4 + a-6z6 + 6a-5z3 - 10a-5z5 + 4a-5z7 - 6a-4z2 + 16a-4z4 - 18a-4z6 + 7a-4z8 + 10a-3z3 - 7a-3z5 - 7a-3z7 + 6a-3z9 + a-2z-2 - 20a-2z2 + 48a-2z4 - 46a-2z6 + 14a-2z8 + 2a-2z10 - 2a-1z-1 + 5a-1z3 + 3a-1z5 - 20a-1z7 + 13a-1z9 + 2z-2 + 1 - 20z2 + 43z4 - 45z6 + 17z8 + 2z10 - 2az-1 + 7az3 - 12az5 - az7 + 7az9 + a2z-2 - 4a2z2 + 8a2z4 - 14a2z6 + 10a2z8 + 5a3z3 - 11a3z5 + 8a3z7 + 2a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   5 1   j = 7                 9 3     j = 5               12 5       j = 3             10 9         j = 1           14 12           j = -1         10 14             j = -3       6 10               j = -5     3 10                 j = -7   1 6                   j = -9   3                     j = -11 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, 434]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 434]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 12, 17, 11], X[8, 4, 9, 3], X[2, 18, 3, 17], > X[14, 6, 15, 5], X[18, 7, 19, 8], X[12, 16, 5, 15], X[20, 14, 21, 13], > X[22, 9, 13, 10], X[10, 21, 11, 22], X[4, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {5, -1, 6, -3, 9, -10, 2, -7}, > {8, -5, 7, -2, 4, -6, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 9 16 20 2 3 4 5 6 24 - q + -- - -- + -- - -- - 22 q + 21 q - 14 q + 8 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 2 4 3 2 9 2 4 6 8 10 2 - q + q + --- - --- + -- + -- + -- + 8 q + 3 q + q + 5 q - 5 q + 12 10 8 6 2 q q q q q 12 14 16 18 > q + q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 434]][a, z]
Out[8]=	2 2 2 4 4 -2 1 a 2 z 3 z 2 2 4 2 z 2 z 2 4 6 -- + ----- + -- + 2 z + -- - ---- + a z - a z + -- - ---- + 2 a z - z - 2 2 2 2 4 2 4 2 z a z z a a a a 6 z > -- 2 a
In[9]:=	Kauffman[Link[11, Alternating, 434]][a, z]
Out[9]=	2 2 2 2 1 a 2 2 a 2 6 z 20 z 2 2 4 2 1 + -- + ----- + -- - --- - --- - 20 z - ---- - ----- - 4 a z + 2 a z + 2 2 2 2 a z z 4 2 z a z z a a 3 3 3 4 4 6 z 10 z 5 z 3 3 3 5 3 4 2 z 16 z > ---- + ----- + ---- + 7 a z + 5 a z - a z + 43 z - ---- + ----- + 5 3 a 6 4 a a a a 4 5 5 5 48 z 2 4 4 4 10 z 7 z 3 z 5 3 5 > ----- + 8 a z - 5 a z - ----- - ---- + ---- - 12 a z - 11 a z + 2 5 3 a a a a 6 6 6 7 7 5 5 6 z 18 z 46 z 2 6 4 6 4 z 7 z > a z - 45 z + -- - ----- - ----- - 14 a z + 4 a z + ---- - ---- - 6 4 2 5 3 a a a a a 7 8 8 9 9 20 z 7 3 7 8 7 z 14 z 2 8 6 z 13 z > ----- - a z + 8 a z + 17 z + ---- + ----- + 10 a z + ---- + ----- + a 4 2 3 a a a a 10 9 10 2 z > 7 a z + 2 z + ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 3 1 6 3 10 6 10 -- + 14 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 10 3 3 2 5 2 5 3 7 3 7 4 > --- + 12 q t + 10 q t + 9 q t + 12 q t + 5 q t + 9 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a434

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