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

K11a217 K11a219

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 16 20 14 2 8 22 10 6

Alexander Polynomial:

t-3 - 10t-2 + 32t-1 - 45 + 32t - 10t2 + t3

Conway Polynomial:

1 + z2 - 4z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{131, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 13q-5 - 18q-4 + 21q-3 - 21q-2 + 19q-1 - 14 + 9q - 4q2 + q3

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

{K11a131, ...}

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 - q-20 - q-18 + 5q-16 - 3q-14 + q-12 + q-10 - 3q-8 + 3q-6 - 4q-4 + 4q-2 + 1 - 2q2 + 4q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + 1 - z2 - 2z4 + a2z2 + a2z4 + a2z6 - a4 - 3a4z2 - 3a4z4 + 2a6 + 3a6z2 - a8

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 5a-1z3 - 9a-1z5 + 4a-1z7 + 1 - 3z2 + 10z4 - 16z6 + 7z8 + az - 2az5 - 8az7 + 6az9 - 6a2z2 + 20a2z4 - 29a2z6 + 10a2z8 + 2a2z10 + a3z - 3a3z3 + 8a3z5 - 18a3z7 + 11a3z9 - a4 + a4z2 + 11a4z4 - 20a4z6 + 9a4z8 + 2a4z10 - a5z + 6a5z3 - 6a5z5 - a5z7 + 5a5z9 - 2a6 + 6a6z2 - 2a6z4 - 5a6z6 + 6a6z8 + 2a7z3 - 6a7z5 + 5a7z7 - a8 + 3a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{1, -3}

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=-2 is the signature of 11218. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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                     3   j = 3                   6 1   j = 1                 8 3     j = -1               11 6       j = -3             11 9         j = -5           10 10           j = -7         8 11             j = -9       5 10               j = -11     2 8                 j = -13   1 5                   j = -15   2                     j = -17 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, 218]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 218]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[16, 8, 17, 7], > X[20, 9, 21, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[8, 16, 9, 15], > X[22, 17, 1, 18], X[10, 19, 11, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 218]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -8, 5, -10, 6, -2, 7, -6, 8, -4, 9, -3, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 218]]
Out[5]=	DTCode[4, 12, 18, 16, 20, 14, 2, 8, 22, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 218]][t]
Out[6]=	-3 10 32 2 3 -45 + t - -- + -- + 32 t - 10 t + t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 218]][z]
Out[7]=	2 4 6 1 + z - 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 218]}
In[9]:=	{KnotDet[Knot[11, Alternating, 218]], KnotSignature[Knot[11, Alternating, 218]]}
Out[9]=	{131, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 218]][q]
Out[10]=	-8 3 7 13 18 21 21 19 2 3 -14 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 131], Knot[11, Alternating, 218]}
In[12]:=	A2Invariant[Knot[11, Alternating, 218]][q]
Out[12]=	-26 -24 2 -20 -18 5 3 -12 -10 3 3 4 1 - q - q + --- - q - q + --- - --- + q + q - -- + -- - -- + 22 16 14 8 6 4 q q q q q q 4 2 4 6 8 10 > -- - 2 q + 4 q - 2 q - q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 218]][a, z]
Out[13]=	2 4 6 8 2 z 2 2 4 2 6 2 4 2 4 1 - a + 2 a - a - z + -- + a z - 3 a z + 3 a z - 2 z + a z - 2 a 4 4 2 6 > 3 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 218]][a, z]
Out[14]=	2 4 6 8 3 5 9 2 z 2 2 4 2 1 - a - 2 a - a + a z + a z - a z + a z - 3 z + -- - 6 a z + a z + 2 a 3 6 2 8 2 5 z 3 3 5 3 7 3 9 3 4 > 6 a z + 3 a z + ---- - 3 a z + 6 a z + 2 a z - 2 a z + 10 z - a 4 5 2 z 2 4 4 4 6 4 8 4 9 z 5 3 5 > ---- + 20 a z + 11 a z - 2 a z - 5 a z - ---- - 2 a z + 8 a z - 2 a a 6 5 5 7 5 9 5 6 z 2 6 4 6 6 6 > 6 a z - 6 a z + a z - 16 z + -- - 29 a z - 20 a z - 5 a z + 2 a 7 8 6 4 z 7 3 7 5 7 7 7 8 2 8 > 3 a z + ---- - 8 a z - 18 a z - a z + 5 a z + 7 z + 10 a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 9 a z + 6 a z + 6 a z + 11 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 218]], Vassiliev[3][Knot[11, Alternating, 218]]}
Out[15]=	{1, -3}
In[16]:=	Kh[Knot[11, Alternating, 218]][q, t]
Out[16]=	9 11 1 2 1 5 2 8 5 10 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 8 11 10 10 11 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a218

K11a217 K11a219