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L11a348 L11a350

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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X14,6,15,5 X18,8,19,7 X20,9,21,10 X10,11,1,12 X6,14,7,13 X4,18,5,17 X22,15,11,16 X2,19,3,20 X16,21,17,22

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 10q-9/2 + 18q-7/2 - 24q-5/2 + 27q-3/2 - 28q-1/2 + 23q1/2 - 18q3/2 + 10q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 4q-14 - 5q-12 + 2q-10 + q-8 - 4q-6 + 6q-4 - 3q-2 + 7 + q2 - q4 + 5q6 - 4q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 6a-3z3 + 8a-3z5 - 4a-3z7 + 3a-2z2 - 11a-2z4 + 16a-2z6 - 8a-2z8 - a-1z-1 + 2a-1z - 8a-1z3 + 8a-1z5 + 8a-1z7 - 8a-1z9 + 1 + 11z2 - 38z4 + 50z6 - 17z8 - 3z10 - az-1 - 5az3 + 26az7 - 18az9 + 14a2z2 - 46a2z4 + 58a2z6 - 22a2z8 - 3a2z10 + 2a3z - 12a3z3 + 13a3z5 + 5a3z7 - 10a3z9 + 6a4z2 - 17a4z4 + 21a4z6 - 13a4z8 + 2a5z - 8a5z3 + 12a5z5 - 9a5z7 - a6z2 + 4a6z4 - 4a6z6 + a7z3 - 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 r = 5 j = 10                       1 j = 8                     3   j = 6                   7 1   j = 4                 11 3     j = 2               12 7       j = 0             16 11         j = -2           13 14           j = -4         11 14             j = -6       7 13               j = -8     3 11                 j = -10   1 7                   j = -12   3                     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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 349]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 349]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[14, 6, 15, 5], X[18, 8, 19, 7], > X[20, 9, 21, 10], X[10, 11, 1, 12], X[6, 14, 7, 13], X[4, 18, 5, 17], > X[22, 15, 11, 16], X[2, 19, 3, 20], X[16, 21, 17, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, > {6, -1, 7, -3, 9, -11, 8, -4, 10, -5, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 10 18 24 27 28 -q + ----- - ---- + ---- - ---- + ---- - ------- + 23 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 9/2 > 18 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 4 5 2 -8 4 6 3 2 4 7 + q - q - q + --- - --- + --- + q - -- + -- - -- + q - q + 14 12 10 6 4 2 q q q q q q 6 8 10 12 14 > 5 q - 4 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 349]][a, z]
Out[8]=	3 3 5 1 a z z 3 5 z 3 z 3 3 3 5 3 2 z -(---) + - + -- - - - a z + a z + -- - ---- + a z - 3 a z + a z - ---- + a z z 3 a 3 a a a a 5 3 5 7 > 2 a z - 2 a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 349]][a, z]
Out[9]=	2 2 1 a 2 z 2 z 3 5 2 z 3 z 2 2 1 - --- - - + --- + --- + 2 a z + 2 a z + 11 z - -- + ---- + 14 a z + a z z 3 a 4 2 a a a 3 3 4 2 6 2 6 z 8 z 3 3 3 5 3 7 3 > 6 a z - a z - ---- - ---- - 5 a z - 12 a z - 8 a z + a z - 3 a a 4 4 5 5 4 2 z 11 z 2 4 4 4 6 4 8 z 8 z > 38 z + ---- - ----- - 46 a z - 17 a z + 4 a z + ---- + ---- + 4 2 3 a a a a 6 6 3 5 5 5 7 5 6 z 16 z 2 6 4 6 > 13 a z + 12 a z - a z + 50 z - -- + ----- + 58 a z + 21 a z - 4 2 a a 7 7 8 6 6 4 z 8 z 7 3 7 5 7 8 8 z > 4 a z - ---- + ---- + 26 a z + 5 a z - 9 a z - 17 z - ---- - 3 a 2 a a 9 2 8 4 8 8 z 9 3 9 10 2 10 > 22 a z - 13 a z - ---- - 18 a z - 10 a z - 3 z - 3 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	14 1 3 1 7 3 11 7 13 16 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 11 14 13 2 2 2 4 2 4 3 > ----- + ---- + ---- + 11 t + 12 q t + 7 q t + 11 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a349

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