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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 5q-7/2 + 10q-5/2 - 16q-3/2 + 21q-1/2 - 25q1/2 + 24q3/2 - 21q5/2 + 15q7/2 - 9q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + 2q-12 + 2q-10 - 2q-8 + 5q-6 - 2q-4 - q-2 + 3 - 4q2 + 5q4 - 2q6 + 3q8 + 3q10 - 4q12 + 3q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 - a-3z-1 + 2a-3z + 4a-3z3 + 2a-3z5 + 2a-1z-1 - 4a-1z3 - 3a-1z5 - a-1z7 - 2az-1 - az + 3az3 + 2az5 + a3z-1 - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 5a-6z4 - 4a-6z6 + 2a-5z - 8a-5z3 + 12a-5z5 - 8a-5z7 - a-4z2 - a-4z4 + 9a-4z6 - 9a-4z8 + a-3z-1 + 3a-3z - 18a-3z3 + 25a-3z5 - 6a-3z7 - 6a-3z9 + 2a-2z2 - 19a-2z4 + 36a-2z6 - 17a-2z8 - 2a-2z10 + 2a-1z-1 - 2a-1z - 14a-1z3 + 16a-1z5 + 12a-1z7 - 13a-1z9 + 1 + z2 - 25z4 + 44z6 - 17z8 - 2z10 + 2az-1 - 4az - 9az3 + 14az5 + 5az7 - 7az9 - 11a2z4 + 20a2z6 - 9a2z8 + a3z-1 - a3z - 4a3z3 + 10a3z5 - 5a3z7 + a4z4 - a4z6

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 = 14                       1 j = 12                     3   j = 10                   6 1   j = 8                 9 3     j = 6               12 6       j = 4             12 9         j = 2           13 12           j = 0         10 14             j = -2       6 11               j = -4     4 10                 j = -6   1 6                   j = -8   4                     j = -10 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 6, 11, 5], > X[8, 4, 9, 3], X[20, 12, 21, 11], X[22, 17, 5, 18], X[18, 21, 19, 22], > X[12, 20, 13, 19], X[16, 10, 17, 9], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 11, -2, 3, -10, 7, -8, > 9, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 5 10 16 21 3/2 5/2 q - ---- + ---- - ---- + ------- - 25 Sqrt[q] + 24 q - 21 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 15 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 2 5 2 -2 2 4 6 8 10 3 - q + --- + --- - -- + -- - -- - q - 4 q + 5 q - 2 q + 3 q + 3 q - 12 10 8 6 4 q q q q q 12 14 16 18 20 > 4 q + 3 q - q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 1]][a, z]
Out[8]=	3 3 3 3 1 2 2 a a z 2 z z 4 z 4 z 3 3 3 -(----) + --- - --- + -- - -- + --- - a z - -- + ---- - ---- + 3 a z - a z + 3 a z z z 5 3 5 3 a a z a a a a 5 5 7 2 z 3 z 5 z > ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 1]][a, z]
Out[9]=	3 2 2 1 2 2 a a 2 z 3 z 2 z 3 2 2 z z 1 + ---- + --- + --- + -- + --- + --- - --- - 4 a z - a z + z - ---- - -- + 3 a z z z 5 3 a 6 4 a z a a a a 2 3 3 3 3 4 4 2 z z 8 z 18 z 14 z 3 3 3 4 5 z z > ---- + -- - ---- - ----- - ----- - 9 a z - 4 a z - 25 z + ---- - -- - 2 7 5 3 a 6 4 a a a a a a 4 5 5 5 5 19 z 2 4 4 4 z 12 z 25 z 16 z 5 > ----- - 11 a z + a z - -- + ----- + ----- + ----- + 14 a z + 2 7 5 3 a a a a a 6 6 6 7 7 3 5 6 4 z 9 z 36 z 2 6 4 6 8 z 6 z > 10 a z + 44 z - ---- + ---- + ----- + 20 a z - a z - ---- - ---- + 6 4 2 5 3 a a a a a 7 8 8 9 9 12 z 7 3 7 8 9 z 17 z 2 8 6 z 13 z > ----- + 5 a z - 5 a z - 17 z - ---- - ----- - 9 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]=	2 1 4 1 6 4 10 6 10 14 + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + -- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 11 2 4 4 2 6 2 6 3 8 3 > ---- + 12 q t + 12 q t + 9 q t + 12 q t + 6 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 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 L11a1

L10n113 L11a2