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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,9,17,10 X14,8,15,7 X18,12,19,11 X20,15,21,16 X22,18,11,17 X10,19,5,20 X8,22,9,21 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-4 + 4q-3 - 8q-2 + 15q-1 - 18 + 23q - 22q2 + 20q3 - 15q4 + 9q5 - 4q6 + q7

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 + 7q-4 + 2q-2 + 7 + 6q2 + 6q6 - 4q8 + 3q10 - q12 - 2q14 + 3q16 - 2q18 + q20

HOMFLY-PT Polynomial:

a-4 + 3a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 - 2a-2 - 9a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 - 2z-2 + 1 + 7z2 + 7z4 + 2z6 + a2z-2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 9a-6z6 - 2a-5z + 10a-5z3 - 19a-5z5 + 14a-5z7 + 2a-4 - 7a-4z2 + 15a-4z4 - 25a-4z6 + 15a-4z8 - 7a-3z + 27a-3z3 - 30a-3z5 - 2a-3z7 + 9a-3z9 + a-2z-2 + 4a-2 - 30a-2z2 + 73a-2z4 - 77a-2z6 + 23a-2z8 + 2a-2z10 - 2a-1z-1 - 7a-1z + 26a-1z3 - 6a-1z5 - 28a-1z7 + 14a-1z9 + 2z-2 + 3 - 29z2 + 67z4 - 57z6 + 12z8 + 2z10 - 2az-1 - 3az + 13az3 - 2az5 - 11az7 + 5az9 + a2z-2 - 8a2z2 + 17a2z4 - 14a2z6 + 4a2z8 - a3z + 3a3z3 - 3a3z5 + a3z7

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 = 15                       1 j = 13                     3   j = 11                   6 1   j = 9                 9 3     j = 7               11 6       j = 5             12 10         j = 3           11 10           j = 1         9 14             j = -1       6 9               j = -3     3 10                 j = -5   1 5                   j = -7   3                     j = -9 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, 403]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 403]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 9, 17, 10], X[14, 8, 15, 7], > X[18, 12, 19, 11], X[20, 15, 21, 16], X[22, 18, 11, 17], X[10, 19, 5, 20], > X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, > {5, -2, 11, -4, 6, -3, 7, -5, 8, -6, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-4 4 8 15 2 3 4 5 6 7 -18 - q + -- - -- + -- + 23 q - 22 q + 20 q - 15 q + 9 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 7 2 2 6 8 10 12 14 7 - q + q + q + -- + -- + 6 q + 6 q - 4 q + 3 q - q - 2 q + 4 2 q q 16 18 20 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 403]][a, z]
Out[8]=	2 2 2 4 -4 2 2 1 a 2 3 z 9 z 2 2 4 3 z 1 + a - -- - -- + ----- + -- + 7 z + ---- - ---- - 2 a z + 7 z + ---- - 2 2 2 2 2 4 2 4 a z a z z a a a 4 6 6 8 10 z 2 4 6 z 5 z z > ----- - a z + 2 z + -- - ---- - -- 2 4 2 2 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 403]][a, z]
Out[9]=	2 2 4 2 1 a 2 2 a 2 z 7 z 7 z 3 3 + -- + -- + -- + ----- + -- - --- - --- - --- - --- - --- - 3 a z - 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 2 z 7 z 30 z 2 2 z 10 z 27 z 26 z > 29 z + ---- - ---- - ----- - 8 a z - -- + ----- + ----- + ----- + 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 3 3 3 4 z 7 z 15 z 73 z 2 4 4 z > 13 a z + 3 a z + 67 z + -- - ---- + ----- + ----- + 17 a z + ---- - 8 6 4 2 7 a a a a a 5 5 5 6 6 6 19 z 30 z 6 z 5 3 5 6 9 z 25 z 77 z > ----- - ----- - ---- - 2 a z - 3 a z - 57 z + ---- - ----- - ----- - 5 3 a 6 4 2 a a a a a 7 7 7 8 8 2 6 14 z 2 z 28 z 7 3 7 8 15 z 23 z > 14 a z + ----- - ---- - ----- - 11 a z + a z + 12 z + ----- + ----- + 5 3 a 4 2 a a a a 9 9 10 2 8 9 z 14 z 9 10 2 z > 4 a z + ---- + ----- + 5 a z + 2 z + ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 5 3 10 6 9 14 q + 11 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 9 q 3 5 5 2 7 2 7 3 9 3 > --- + 10 q t + 12 q t + 10 q t + 11 q t + 6 q t + 9 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a403

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