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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,8,19,7 X16,6,17,5 X10,16,11,15 X14,3,15,4 X4,11,5,12 X2,20,3,19 X20,9,21,10 X12,13,7,14 X22,18,13,17 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

q-6 - 6q-5 + 15q-4 - 23q-3 + 33q-2 - 36q-1 + 37 - 31q + 23q2 - 13q3 + 5q4 - q5

A2 (sl(3)) Invariant:

q-18 - 3q-16 + q-14 + 5q-12 - 3q-10 + 11q-8 + 3q-6 + 4q-4 + 8q-2 - 5 + 8q2 - 6q4 + 2q6 + 4q8 - 5q10 + 3q12 - q14

HOMFLY-PT Polynomial:

- 2a-2z2 - 2a-2z4 - a-2z6 + z-2 + 1 + 4z2 + 7z4 + 4z6 + z8 - 2a2z-2 - a2 - 2a2z2 - 4a2z4 - 2a2z6 + a4z-2 + a4z4

Kauffman Polynomial:

a-5z5 - 2a-4z4 + 5a-4z6 + 5a-3z3 - 13a-3z5 + 13a-3z7 - 6a-2z2 + 21a-2z4 - 32a-2z6 + 21a-2z8 + 3a-1z3 - 21a-1z7 + 19a-1z9 - z-2 + 1 - 10z2 + 53z4 - 81z6 + 28z8 + 7z10 + 2az-1 - az - az3 + 26az5 - 67az7 + 36az9 - 2a2z-2 + a2 - 2a2z2 + 41a2z4 - 72a2z6 + 22a2z8 + 7a2z10 + 2a3z-1 - a3z + a3z3 + 6a3z5 - 27a3z7 + 17a3z9 - a4z-2 + a4 + 2a4z2 + 11a4z4 - 27a4z6 + 15a4z8 - 6a5z5 + 6a5z7 + a6z6

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 r = 5 j = 11                       1 j = 9                     4   j = 7                   9 1   j = 5                 14 4     j = 3               17 9       j = 1             20 14         j = -1           18 19           j = -3         15 18             j = -5       10 20               j = -7     5 13                 j = -9   1 10                   j = -11   5                     j = -13 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, 520]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 520]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 8, 19, 7], X[16, 6, 17, 5], X[10, 16, 11, 15], > X[14, 3, 15, 4], X[4, 11, 5, 12], X[2, 20, 3, 19], X[20, 9, 21, 10], > X[12, 13, 7, 14], X[22, 18, 13, 17], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 5, -6, 3, -11}, {2, -1, 8, -4, 6, -9}, > {9, -5, 4, -3, 10, -2, 7, -8, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-6 6 15 23 33 36 2 3 4 5 37 + q - -- + -- - -- + -- - -- - 31 q + 23 q - 13 q + 5 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 -14 5 3 11 3 4 8 2 4 6 -5 + q - --- + q + --- - --- + -- + -- + -- + -- + 8 q - 6 q + 2 q + 16 12 10 8 6 4 2 q q q q q q q 8 10 12 14 > 4 q - 5 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 520]][a, z]
Out[8]=	2 4 2 4 2 -2 2 a a 2 2 z 2 2 4 2 z 2 4 1 - a + z - ---- + -- + 4 z - ---- - 2 a z + 7 z - ---- - 4 a z + 2 2 2 2 z z a a 6 4 4 6 z 2 6 8 > a z + 4 z - -- - 2 a z + z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 520]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 6 z 1 + a + a - z - ---- - -- + --- + ---- - a z - a z - 10 z - ---- - 2 2 z z 2 z z a 3 3 4 4 2 2 4 2 5 z 3 z 3 3 3 4 2 z 21 z > 2 a z + 2 a z + ---- + ---- - a z + a z + 53 z - ---- + ----- + 3 a 4 2 a a a 5 5 2 4 4 4 z 13 z 5 3 5 5 5 6 > 41 a z + 11 a z + -- - ----- + 26 a z + 6 a z - 6 a z - 81 z + 5 3 a a 6 6 7 7 5 z 32 z 2 6 4 6 6 6 13 z 21 z 7 > ---- - ----- - 72 a z - 27 a z + a z + ----- - ----- - 67 a z - 4 2 3 a a a a 8 9 3 7 5 7 8 21 z 2 8 4 8 19 z > 27 a z + 6 a z + 28 z + ----- + 22 a z + 15 a z + ----- + 2 a a 9 3 9 10 2 10 > 36 a z + 17 a z + 7 z + 7 a z
In[10]:=	Kh[L][q, t]
Out[10]=	19 1 5 1 10 5 13 10 20 -- + 20 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 15 18 18 3 3 2 5 2 5 3 > ----- + ---- + --- + 14 q t + 17 q t + 9 q t + 14 q t + 4 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 9 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a520

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