© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a79 K11a81

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X22,10,1,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X6,17,7,18 X8,20,9,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 22 2 18 20 6 8 16

Alexander Polynomial:

t-4 - 6t-3 + 16t-2 - 28t-1 + 35 - 28t + 16t2 - 6t3 + t4

Conway Polynomial:

1 - 2z2 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{137, 0}

Jones Polynomial:

q-6 - 4q-5 + 9q-4 - 14q-3 + 19q-2 - 22q-1 + 22 - 19q + 14q2 - 8q3 + 4q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a270, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-2 - 2a-2z2 - 3a-2z4 - a-2z6 + 5z2 + 9z4 + 5z6 + z8 - a2 - 7a2z2 - 7a2z4 - 2a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 - a-3z + 4a-3z3 - 10a-3z5 + 7a-3z7 - a-2 + 3a-2z2 - 9a-2z6 + 8a-2z8 - 4a-1z + 13a-1z3 - 12a-1z5 - a-1z7 + 6a-1z9 - 5z2 + 27z4 - 34z6 + 13z8 + 2z10 - 4az + 11az3 + az5 - 19az7 + 12az9 + a2 - 11a2z2 + 33a2z4 - 38a2z6 + 12a2z8 + 2a2z10 - 2a3z + 8a3z3 - 7a3z5 - 7a3z7 + 6a3z9 + a4 - 4a4z2 + 10a4z4 - 16a4z6 + 7a4z8 - a5z + 5a5z3 - 9a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-2, 1}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 1180. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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 = 11                       1 j = 9                     3   j = 7                   5 1   j = 5                 9 3     j = 3               10 5       j = 1             12 9         j = -1           11 11           j = -3         8 11             j = -5       6 11               j = -7     3 8                 j = -9   1 6                   j = -11   3                     j = -13 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 80]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 80]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[22, 10, 1, 9], X[2, 11, 3, 12], X[18, 14, 19, 13], X[20, 15, 21, 16], > X[6, 17, 7, 18], X[8, 20, 9, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 80]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, > -8, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 80]]
Out[5]=	DTCode[4, 10, 12, 14, 22, 2, 18, 20, 6, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 80]][t]
Out[6]=	-4 6 16 28 2 3 4 35 + t - -- + -- - -- - 28 t + 16 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 80]][z]
Out[7]=	2 6 8 1 - 2 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 80]}
In[9]:=	{KnotDet[Knot[11, Alternating, 80]], KnotSignature[Knot[11, Alternating, 80]]}
Out[9]=	{137, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 80]][q]
Out[10]=	-6 4 9 14 19 22 2 3 4 5 22 + q - -- + -- - -- + -- - -- - 19 q + 14 q - 8 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 80], Knot[11, Alternating, 270]}
In[12]:=	A2Invariant[Knot[11, Alternating, 80]][q]
Out[12]=	-18 -16 2 3 4 -6 -4 2 2 4 6 -5 + q - q + --- - --- + -- - q - q + -- + 4 q - 3 q + 2 q + 12 10 8 2 q q q q 8 10 12 14 > 3 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 80]][a, z]
Out[13]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 3 z 2 4 a - a + a + 5 z - ---- - 7 a z + 2 a z + 9 z - ---- - 7 a z + 2 2 a a 6 4 4 6 z 2 6 8 > a z + 5 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 80]][a, z]
Out[14]=	2 2 -2 2 4 z 4 z 3 5 2 2 z 3 z -a + a + a - -- - --- - 4 a z - 2 a z - a z - 5 z + ---- + ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 6 2 z 4 z 13 z 3 3 3 > 11 a z - 4 a z + a z - -- + ---- + ----- + 11 a z + 8 a z + 5 3 a a a 4 5 5 5 3 4 6 z 2 4 4 4 6 4 z 10 z > 5 a z + 27 z - ---- + 33 a z + 10 a z - 2 a z + -- - ----- - 4 5 3 a a a 5 6 6 12 z 5 3 5 5 5 6 4 z 9 z 2 6 > ----- + a z - 7 a z - 9 a z - 34 z + ---- - ---- - 38 a z - a 4 2 a a 7 7 8 4 6 6 6 7 z z 7 3 7 5 7 8 8 z > 16 a z + a z + ---- - -- - 19 a z - 7 a z + 4 a z + 13 z + ---- + 3 a 2 a a 9 2 8 4 8 6 z 9 3 9 10 2 10 > 12 a z + 7 a z + ---- + 12 a z + 6 a z + 2 z + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 80]], Vassiliev[3][Knot[11, Alternating, 80]]}
Out[15]=	{-2, 1}
In[16]:=	Kh[Knot[11, Alternating, 80]][q, t]
Out[16]=	11 1 3 1 6 3 8 6 11 -- + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 8 11 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 10 q t + 5 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a80

K11a79 K11a81