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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 5q-11/2 - 11q-9/2 + 15q-7/2 - 22q-5/2 + 20q-3/2 - 22q-1/2 + 15q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - 2q-18 - q-16 + 5q-14 + q-12 + 12q-10 + 13q-8 + 13q-6 + 18q-4 + 7q-2 + 12 + q2 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z-3 - a-1z-1 - a-1z3 - a-1z5 + 3az-3 + 2az-1 + 3az3 + 3az5 + az7 - 3a3z-3 - a3z-1 - 3a3z3 - 2a3z5 + a5z-3 + a5z3

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 + a-1z-3 - a-1z-1 - 8a-1z3 + 17a-1z5 - 11a-1z7 - 3z-2 + 2 + 12z6 - 11z8 + 3az-3 - az - 24az3 + 48az5 - 19az7 - 4az9 - 6a2z-2 + 3a2 - 8a2z4 + 34a2z6 - 22a2z8 + 3a3z-3 - a3z - 24a3z3 + 48a3z5 - 19a3z7 - 4a3z9 - 3a4z-2 + 2a4 + 12a4z6 - 11a4z8 + a5z-3 - a5z-1 - 8a5z3 + 17a5z5 - 11a5z7 + 4a6z4 - 5a6z6 - a7z5

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 j = 8                     1 j = 6                   4   j = 4                 7 1   j = 2               8 4     j = 0             14 7       j = -2           10 12         j = -4         12 10           j = -6       7 14             j = -8     4 8               j = -10   1 7                 j = -12   4                   j = -14 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 169]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 169]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[14, 9, 11, 10], X[20, 13, 15, 14], X[10, 19, 5, 20], > X[18, 12, 19, 11], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 6, -8}, > {4, -5, 10, -9, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 5 11 15 22 20 22 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 5 -12 12 13 13 18 7 2 6 12 + q - --- - q + --- + q + --- + -- + -- + -- + -- + q + 3 q - 18 14 10 8 6 4 2 q q q q q q q 8 10 > 3 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 169]][a, z]
Out[8]=	3 5 3 3 1 3 a 3 a a 1 2 a a z 3 3 3 5 3 -(----) + --- - ---- + -- - --- + --- - -- - -- + 3 a z - 3 a z + a z - 3 3 3 3 a z z z a a z z z z 5 z 5 3 5 7 > -- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 169]][a, z]
Out[9]=	3 5 2 4 5 2 4 1 3 a 3 a a 3 6 a 3 a 1 a 2 + 3 a + 2 a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - -- - a z - 3 3 3 3 2 2 2 a z z a z z z z z z z 3 4 3 8 z 3 3 3 5 3 4 z 2 4 6 4 > a z - ---- - 24 a z - 24 a z - 8 a z + ---- - 8 a z + 4 a z - a 2 a 5 5 6 z 17 z 5 3 5 5 5 7 5 6 5 z > -- + ----- + 48 a z + 48 a z + 17 a z - a z + 12 z - ---- + 3 a 2 a a 7 2 6 4 6 6 6 11 z 7 3 7 5 7 > 34 a z + 12 a z - 5 a z - ----- - 19 a z - 19 a z - 11 a z - a 8 2 8 4 8 9 3 9 > 11 z - 22 a z - 11 a z - 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	12 1 4 1 7 4 8 7 14 14 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 12 10 10 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 7 t + 8 q t + 4 q t + 7 q t + q t + 4 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a169

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