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

K11a95 K11a97

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 18 14 2 22 8 20 6 16

Alexander Polynomial:

- t-3 + 9t-2 - 29t-1 + 43 - 29t + 9t2 - t3

Conway Polynomial:

1 - 2z2 + 3z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 12q-3 + 16q-2 - 19q-1 + 20 - 17q + 13q2 - 8q3 + 4q4 - q5

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

{K11a28, K11a87, ...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 + q-12 - 4q-10 + 3q-8 - q-6 - q-4 + 3q-2 - 2 + 4q2 - 3q4 + 3q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2 + 2a-2z2 + 2a-2z4 - 1 - 4z2 - 2z4 - z6 + 2a2 + 4a2z2 + 3a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 + 5a-3z3 - 11a-3z5 + 7a-3z7 - a-2 + 3a-2z2 - a-2z4 - 7a-2z6 + 7a-2z8 + 8a-1z3 - 16a-1z5 + 5a-1z7 + 4a-1z9 - 1 + 3z2 + 7z4 - 21z6 + 12z8 + z10 + 6az3 - 11az5 - 3az7 + 7az9 - 2a2 + 6a2z2 + 2a2z4 - 17a2z6 + 9a2z8 + a2z10 - 2a3z + 11a3z3 - 15a3z5 + 2a3z7 + 3a3z9 - 2a4 + 7a4z2 - 3a4z4 - 6a4z6 + 4a4z8 - 2a5z + 7a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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

{-2, 2}

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 1196. 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                 8 3     j = 3               9 5       j = 1             11 8         j = -1           9 10           j = -3         7 10             j = -5       5 9               j = -7     2 7                 j = -9   1 5                   j = -11   2                     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, 96]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 96]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[18, 7, 19, 8], > X[14, 9, 15, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[8, 15, 9, 16], > X[20, 18, 21, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 96]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -5, 8, -11, 9, -4, 10, > -9, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 96]]
Out[5]=	DTCode[4, 10, 12, 18, 14, 2, 22, 8, 20, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 96]][t]
Out[6]=	-3 9 29 2 3 43 - t + -- - -- - 29 t + 9 t - t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 96]][z]
Out[7]=	2 4 6 1 - 2 z + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 96]}
In[9]:=	{KnotDet[Knot[11, Alternating, 96]], KnotSignature[Knot[11, Alternating, 96]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 96]][q]
Out[10]=	-6 3 7 12 16 19 2 3 4 5 20 + q - -- + -- - -- + -- - -- - 17 q + 13 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, 28], Knot[11, Alternating, 87], > Knot[11, Alternating, 96]}
In[12]:=	A2Invariant[Knot[11, Alternating, 96]][q]
Out[12]=	-20 -18 2 -14 -12 4 3 -6 -4 3 2 -2 + q + q - --- + q + q - --- + -- - q - q + -- + 4 q - 16 10 8 2 q q q q 4 8 10 12 14 16 > 3 q + 3 q - 3 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 96]][a, z]
Out[13]=	2 2 -2 2 4 6 2 z 2 z 2 2 4 2 4 -1 + a + 2 a - 2 a + a - 4 z - -- + ---- + 4 a z - 3 a z - 2 z + 4 2 a a 4 2 z 2 4 6 > ---- + 3 a z - z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 96]][a, z]
Out[14]=	2 2 -2 2 4 6 3 5 2 2 z 3 z 2 2 -1 - a - 2 a - 2 a - a - 2 a z - 2 a z + 3 z + ---- + ---- + 6 a z + 4 2 a a 3 3 3 4 2 6 2 z 5 z 8 z 3 3 3 5 3 4 > 7 a z + 3 a z - -- + ---- + ---- + 6 a z + 11 a z + 7 a z + 7 z - 5 3 a a a 4 4 5 5 5 6 z z 2 4 4 4 6 4 z 11 z 16 z 5 > ---- - -- + 2 a z - 3 a z - 3 a z + -- - ----- - ----- - 11 a z - 4 2 5 3 a a a a a 6 6 3 5 5 5 6 4 z 7 z 2 6 4 6 6 6 > 15 a z - 8 a z - 21 z + ---- - ---- - 17 a z - 6 a z + a z + 4 2 a a 7 7 8 7 z 5 z 7 3 7 5 7 8 7 z 2 8 > ---- + ---- - 3 a z + 2 a z + 3 a z + 12 z + ---- + 9 a z + 3 a 2 a a 9 4 8 4 z 9 3 9 10 2 10 > 4 a z + ---- + 7 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 96]], Vassiliev[3][Knot[11, Alternating, 96]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, Alternating, 96]][q, t]
Out[16]=	10 1 2 1 5 2 7 5 9 -- + 11 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 7 10 9 3 3 2 5 2 5 3 > ----- + ---- + --- + 8 q t + 9 q t + 5 q t + 8 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 K11a96

K11a95 K11a97