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L11a267 L11a269

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 8q-11/2 - 14q-9/2 + 18q-7/2 - 21q-5/2 + 21q-3/2 - 19q-1/2 + 13q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - q-18 + q-16 + 3q-14 - 3q-12 + 4q-10 + q-6 + 3q-4 - 3q-2 + 5 - 3q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 + 3az + 7az3 + 4az5 + az7 - a3z-1 - 7a3z - 8a3z3 - 3a3z5 + a5z-1 + 4a5z + 3a5z3 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 11a-1z5 - 7a-1z7 - 3z2 + 4z4 + 5z6 - 7z8 + 7az - 29az3 + 33az5 - 10az7 - 4az9 + a2z2 - 17a2z4 + 32a2z6 - 16a2z8 - a2z10 - a3z-1 + 14a3z - 42a3z3 + 40a3z5 - 3a3z7 - 8a3z9 + a4 + 3a4z2 - 22a4z4 + 36a4z6 - 15a4z8 - a4z10 - a5z-1 + 11a5z - 29a5z3 + 29a5z5 - 4a5z7 - 4a5z9 - 5a6z4 + 12a6z6 - 6a6z8 + 2a7z - 8a7z3 + 10a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 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                     3   j = 4                   5 1   j = 2                 8 3     j = 0               11 5       j = -2             11 9         j = -4           10 10           j = -6         8 11             j = -8       6 10               j = -10     3 9                 j = -12   1 5                   j = -14   3                     j = -16 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, 268]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 268]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[16, 14, 17, 13], X[14, 5, 15, 6], X[4, 15, 5, 16], X[6, 22, 7, 21], > X[18, 7, 19, 8], X[20, 17, 21, 18], X[8, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -8, 9, -11}, > {4, -1, 3, -2, 5, -6, 7, -5, 10, -9, 11, -10, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 8 14 18 21 21 19 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 -18 -16 3 3 4 -6 3 3 2 4 5 - q + --- - q + q + --- - --- + --- + q + -- - -- - 3 q - q + 20 14 12 10 4 2 q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 268]][a, z]
Out[8]=	3 5 3 a a z 3 5 7 2 z 3 3 3 -(--) + -- - - + 3 a z - 7 a z + 4 a z - a z - ---- + 7 a z - 8 a z + z z a a 5 5 3 z 5 3 5 7 > 3 a z - -- + 4 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 268]][a, z]
Out[9]=	3 5 2 4 a a 2 z 3 5 7 2 2 z 2 2 a - -- - -- + --- + 7 a z + 14 a z + 11 a z + 2 a z - 3 z - ---- + a z + z z a 2 a 3 3 4 2 8 2 z 7 z 3 3 3 5 3 7 3 > 3 a z - a z + -- - ---- - 29 a z - 42 a z - 29 a z - 8 a z + 3 a a 4 5 5 4 6 z 2 4 4 4 6 4 8 4 z 11 z > 4 z + ---- - 17 a z - 22 a z - 5 a z + 2 a z - -- + ----- + 2 3 a a a 6 5 3 5 5 5 7 5 6 4 z 2 6 > 33 a z + 40 a z + 29 a z + 10 a z + 5 z - ---- + 32 a z + 2 a 7 4 6 6 6 8 6 7 z 7 3 7 5 7 > 36 a z + 12 a z - a z - ---- - 10 a z - 3 a z - 4 a z - a 7 7 8 2 8 4 8 6 8 9 3 9 > 4 a z - 7 z - 16 a z - 15 a z - 6 a z - 4 a z - 8 a z - 5 9 2 10 4 10 > 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 5 3 9 6 10 11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 8 11 10 10 11 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 8 q t + 3 q t + 5 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a268

L11a267 L11a269