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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X22,16,19,15 X20,9,21,10 X8,19,9,20 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X16,22,17,21 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 15q-2 - 19q-1 + 25 - 22q + 20q2 - 15q3 + 9q4 - 4q5 + q6

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 3q-10 + 2q-8 + 2q-6 + 11q-2 + 3 + 9q2 + 2q4 - q6 + 3q8 - 4q10 + 2q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 + a-2z-2 - 3a-2 - 8a-2z2 - 7a-2z4 - 2a-2z6 - 2z-2 + 3 + 9z2 + 10z4 + 5z6 + z8 + a2z-2 - a2 - 3a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 9a-5z5 + 4a-5z7 + 2a-4 - 6a-4z2 + 9a-4z4 - 15a-4z6 + 7a-4z8 - 7a-3z + 19a-3z3 - 17a-3z5 - 4a-3z7 + 6a-3z9 + a-2z-2 + 6a-2 - 30a-2z2 + 60a-2z4 - 55a-2z6 + 16a-2z8 + 2a-2z10 - 2a-1z-1 - 9a-1z + 21a-1z3 - 24a-1z7 + 14a-1z9 + 2z-2 + 7 - 36z2 + 78z4 - 66z6 + 20z8 + 2z10 - 2az-1 - 5az + 13az3 - 5az5 - 8az7 + 8az9 + a2z-2 + 2a2 - 13a2z2 + 24a2z4 - 23a2z6 + 11a2z8 - a3z + 4a3z3 - 12a3z5 + 8a3z7 - 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                   6 1   j = 7                 9 3     j = 5               11 6       j = 3             11 9         j = 1           14 11           j = -1         9 15             j = -3       6 10               j = -5     3 9                 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, 492]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 492]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[22, 16, 19, 15], X[20, 9, 21, 10], > X[8, 19, 9, 20], X[4, 17, 1, 18], X[12, 6, 13, 5], X[10, 4, 11, 3], > X[18, 12, 5, 11], X[16, 22, 17, 21], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 8, -6}, {5, -4, 10, -3}, > {7, -1, 2, -5, 4, -8, 9, -7, 11, -2, 3, -10, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 9 15 19 2 3 4 5 6 25 - q + -- - -- + -- - -- - 22 q + 20 q - 15 q + 9 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 3 2 2 11 2 4 6 8 10 12 3 - q + --- - --- + -- + -- + -- + 9 q + 2 q - q + 3 q - 4 q + 2 q - 12 10 8 6 2 q q q q q 16 18 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 492]][a, z]
Out[8]=	2 2 2 -4 3 2 2 1 a 2 2 z 8 z 2 2 4 3 + a - -- - a - -- + ----- + -- + 9 z + ---- - ---- - 3 a z + 10 z + 2 2 2 2 2 4 2 a z a z z a a 4 4 6 z 7 z 2 4 6 2 z 2 6 8 > -- - ---- - 3 a z + 5 z - ---- - a z + z 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 492]][a, z]
Out[9]=	2 2 6 2 2 1 a 2 2 a 2 z 7 z 9 z 7 + -- + -- + 2 a + -- + ----- + -- - --- - --- - --- - --- - --- - 5 a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 3 2 z 6 z 30 z 2 2 6 z 19 z 21 z > a z - 36 z + -- - ---- - ----- - 13 a z + ---- + ----- + ----- + 6 4 2 5 3 a a a a a a 4 4 4 3 3 3 5 3 4 2 z 9 z 60 z 2 4 > 13 a z + 4 a z - a z + 78 z - ---- + ---- + ----- + 24 a z - 6 4 2 a a a 5 5 6 6 4 4 9 z 17 z 5 3 5 5 5 6 z 15 z > 5 a z - ---- - ----- - 5 a z - 12 a z + a z - 66 z + -- - ----- - 5 3 6 4 a a a a 6 7 7 7 55 z 2 6 4 6 4 z 4 z 24 z 7 3 7 > ----- - 23 a z + 4 a z + ---- - ---- - ----- - 8 a z + 8 a z + 2 5 3 a a a a 8 8 9 9 10 8 7 z 16 z 2 8 6 z 14 z 9 10 2 z > 20 z + ---- + ----- + 11 a z + ---- + ----- + 8 a z + 2 z + ----- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	15 1 3 1 6 3 9 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 9 3 3 2 5 2 5 3 7 3 7 4 > --- + 11 q t + 11 q t + 9 q t + 11 q t + 6 q t + 9 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a492

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