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L11a335 L11a337

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DrawMorseLink

PD Presentation:

X10,1,11,2 X8,9,1,10 X14,3,15,4 X22,11,9,12 X2,15,3,16 X4,19,5,20 X20,5,21,6 X18,21,19,22 X6,13,7,14 X16,7,17,8 X12,17,13,18

Gauss Code:

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

Jones Polynomial:

q-27/2 - 5q-25/2 + 10q-23/2 - 16q-21/2 + 21q-19/2 - 23q-17/2 + 23q-15/2 - 20q-13/2 + 13q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 + 4q-38 - 2q-36 + q-34 + 2q-32 - 5q-30 + 3q-28 - 2q-26 + 2q-24 + 3q-22 - 2q-20 + 6q-18 - 2q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - a7z-1 - 5a7z - 7a7z3 - 3a7z5 + a9z-1 - 2a9z - 6a9z3 - 3a9z5 + 5a11z + 4a11z3 - a13z

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - a7z-1 + 6a7z - 10a7z3 + 10a7z5 - 6a7z7 + a8 - 3a8z2 - a8z4 + 7a8z6 - 7a8z8 - a9z-1 + 3a9z - 9a9z3 + 7a9z5 + 2a9z7 - 6a9z9 + 11a10z2 - 36a10z4 + 40a10z6 - 15a10z8 - 2a10z10 - a11z - 7a11z3 + 3a11z5 + 18a11z7 - 13a11z9 + 15a12z2 - 43a12z4 + 51a12z6 - 17a12z8 - 2a12z10 + 2a13z - 14a13z3 + 17a13z5 + 5a13z7 - 7a13z9 + 2a14z2 - 11a14z4 + 20a14z6 - 9a14z8 - a15z - 4a15z3 + 10a15z5 - 5a15z7 + a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     3 1 j = -8                   5     j = -10                 8 3     j = -12               12 5       j = -14             11 8         j = -16           12 12           j = -18         9 11             j = -20       7 12               j = -22     4 10                 j = -24   1 6                   j = -26   4                     j = -28 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, 336]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 336]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[14, 3, 15, 4], X[22, 11, 9, 12], > X[2, 15, 3, 16], X[4, 19, 5, 20], X[20, 5, 21, 6], X[18, 21, 19, 22], > X[6, 13, 7, 14], X[16, 7, 17, 8], X[12, 17, 13, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, -6, 7, -9, 10, -2}, > {2, -1, 4, -11, 9, -3, 5, -10, 11, -8, 6, -7, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 5 10 16 21 23 23 20 13 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 8 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 4 2 -34 2 5 3 2 2 3 2 6 -q + --- - --- + q + --- - --- + --- - --- + --- + --- - --- + --- - 38 36 32 30 28 26 24 22 20 18 q q q q q q q q q q 2 3 2 -8 > --- + --- - --- + q 16 12 10 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 336]][a, z]
Out[8]=	7 9 a a 5 7 9 11 13 5 3 7 3 -(--) + -- - a z - 5 a z - 2 a z + 5 a z - a z - 2 a z - 7 a z - z z 9 3 11 3 5 5 7 5 9 5 > 6 a z + 4 a z - a z - 3 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 336]][a, z]
Out[9]=	7 9 8 a a 5 7 9 11 13 15 6 2 a - -- - -- - a z + 6 a z + 3 a z - a z + 2 a z - a z - a z - z z 8 2 10 2 12 2 14 2 5 3 7 3 9 3 > 3 a z + 11 a z + 15 a z + 2 a z + 2 a z - 10 a z - 9 a z - 11 3 13 3 15 3 6 4 8 4 10 4 12 4 > 7 a z - 14 a z - 4 a z + 4 a z - a z - 36 a z - 43 a z - 14 4 16 4 5 5 7 5 9 5 11 5 13 5 > 11 a z + a z - a z + 10 a z + 7 a z + 3 a z + 17 a z + 15 5 6 6 8 6 10 6 12 6 14 6 > 10 a z - 3 a z + 7 a z + 40 a z + 51 a z + 20 a z - 16 6 7 7 9 7 11 7 13 7 15 7 8 8 > a z - 6 a z + 2 a z + 18 a z + 5 a z - 5 a z - 7 a z - 10 8 12 8 14 8 9 9 11 9 13 9 > 15 a z - 17 a z - 9 a z - 6 a z - 13 a z - 7 a z - 10 10 12 10 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 4 1 6 4 10 7 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 12 9 11 12 12 11 8 12 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 5 8 3 5 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a336

L11a335 L11a337