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L11n407 L11n409

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,16,8,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-7 + 2q-6 - 3q-5 + 4q-4 - 3q-3 + 3q-2 - q-1 + 2 + q - q2 + q3

A2 (sl(3)) Invariant:

- q-22 - q-16 + q-14 + 2q-10 + 2q-8 + 2q-6 + 4q-4 + 3q-2 + 5 + 4q2 + 2q4 + 2q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 2z-2 - 5 - 5z2 - z4 + a2z-2 + 3a2 + 3a2z2 + a2z4 + a4 + 2a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 5a-2z4 + a-2z6 - 2a-1z-1 + 3a-1z + 3a-1z3 - 5a-1z5 + a-1z7 + 2z-2 - 9 + 19z2 - 10z4 + z6 - 2az-1 + 7az - 3az3 - az5 + a2z-2 - 6a2 + 7a2z2 + 2a2z4 - 5a2z6 + a2z8 + 3a3z - 7a3z3 + 7a3z5 - 5a3z7 + a3z9 + 2a4 - 13a4z2 + 21a4z4 - 15a4z6 + 3a4z8 - 3a5z + 6a5z3 - 2a5z5 - 3a5z7 + a5z9 + 2a6 - 7a6z2 + 14a6z4 - 10a6z6 + 2a6z8 - 2a7z + 7a7z3 - 5a7z5 + a7z7

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 = 7                       1 j = 5                         j = 3                   1 1   j = 1               4 1       j = -1             2 4 1       j = -3           2 2 2         j = -5         2 2             j = -7       2 2 1             j = -9     1 2                 j = -11   1 2                   j = -13   1                     j = -15 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, NonAlternating, 408]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 408]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 16, 8, 17], X[17, 19, 18, 22], > X[11, 20, 12, 21], X[19, 10, 20, 11], X[21, 5, 22, 18], X[9, 14, 10, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-6, 5, -7, 4}, > {10, -1, -3, 9, -8, 6, -5, -2, 11, 8, -9, 3, -4, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 3 4 3 3 1 2 3 2 - q + -- - -- + -- - -- + -- - - + q - q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -16 -14 2 2 2 4 3 2 4 6 8 10 5 - q - q + q + --- + -- + -- + -- + -- + 4 q + 2 q + 2 q + q + q 10 8 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 408]][a, z]
Out[8]=	2 2 2 2 4 6 2 1 a 2 z 2 2 4 2 -5 + -- + 3 a + a - a - -- + ----- + -- - 5 z + -- + 3 a z + 2 a z - 2 2 2 2 2 2 a z a z z a 6 2 4 2 4 4 4 > a z - z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 408]][a, z]
Out[9]=	2 4 2 4 6 2 1 a 2 2 a 3 z -9 - -- - 6 a + 2 a + 2 a + -- + ----- + -- - --- - --- + --- + 7 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 7 2 6 z 2 2 4 2 6 2 > 3 a z - 3 a z - 2 a z + 19 z + ---- + 7 a z - 13 a z - 7 a z + 2 a 3 4 3 z 3 3 3 5 3 7 3 4 5 z 2 4 > ---- - 3 a z - 7 a z + 6 a z + 7 a z - 10 z - ---- + 2 a z + a 2 a 5 6 4 4 6 4 5 z 5 3 5 5 5 7 5 6 z > 21 a z + 14 a z - ---- - a z + 7 a z - 2 a z - 5 a z + z + -- - a 2 a 7 2 6 4 6 6 6 z 3 7 5 7 7 7 2 8 > 5 a z - 15 a z - 10 a z + -- - 5 a z - 3 a z + a z + a z + a 4 8 6 8 3 9 5 9 > 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 1 2 1 2 2 -- + - + 4 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 2 2 1 2 2 2 2 t 3 2 > ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q t + q t + 7 3 5 3 7 2 5 2 3 2 3 q t q q t q t q t q t q t q t 3 3 7 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n408

L11n407 L11n409