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

K11a208 K11a210

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 20 14 18 2 8 22 10 6

Alexander Polynomial:

- t-3 + 10t-2 - 34t-1 + 51 - 34t + 10t2 - t3

Conway Polynomial:

1 - 3z2 + 4z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{141, 0}

Jones Polynomial:

q-6 - 3q-5 + 8q-4 - 14q-3 + 19q-2 - 23q-1 + 23 - 20q + 16q2 - 9q3 + 4q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + 2q-14 + 2q-12 - 5q-10 + 3q-8 - 2q-6 - 2q-4 + 3q-2 - 3 + 5q2 - 3q4 + q6 + 4q8 - 4q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 5a-4z4 + 4a-4z6 + 5a-3z3 - 11a-3z5 + 8a-3z7 - a-2 - a-2z2 + 7a-2z4 - 14a-2z6 + 10a-2z8 + 2a-1z + 2a-1z3 - 8a-1z5 - 2a-1z7 + 7a-1z9 - 9z2 + 26z4 - 35z6 + 15z8 + 2z10 + 6az - 10az3 + 9az5 - 18az7 + 12az9 - 7a2z2 + 20a2z4 - 27a2z6 + 10a2z8 + 2a2z10 + 3a3z - a3z3 - 2a3z5 - 5a3z7 + 5a3z9 - a4 + 2a4z2 + 3a4z4 - 9a4z6 + 5a4z8 - a5z + 5a5z3 - 7a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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

{-3, 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=0 is the signature of 11209. 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                   6 1   j = 5                 10 3     j = 3               10 6       j = 1             13 10         j = -1           11 11           j = -3         8 12             j = -5       6 11               j = -7     2 8                 j = -9   1 6                   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, 209]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 209]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[20, 7, 21, 8], > X[14, 10, 15, 9], X[18, 11, 19, 12], X[2, 13, 3, 14], X[8, 16, 9, 15], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 209]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -8, 5, -10, 6, -2, 7, -5, 8, -3, 9, -6, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 209]]
Out[5]=	DTCode[4, 12, 16, 20, 14, 18, 2, 8, 22, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 209]][t]
Out[6]=	-3 10 34 2 3 51 - t + -- - -- - 34 t + 10 t - t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 209]][z]
Out[7]=	2 4 6 1 - 3 z + 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 209]}
In[9]:=	{KnotDet[Knot[11, Alternating, 209]], KnotSignature[Knot[11, Alternating, 209]]}
Out[9]=	{141, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 209]][q]
Out[10]=	-6 3 8 14 19 23 2 3 4 5 23 + q - -- + -- - -- + -- - -- - 20 q + 16 q - 9 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, 209]}
In[12]:=	A2Invariant[Knot[11, Alternating, 209]][q]
Out[12]=	-20 -18 2 2 2 5 3 2 2 3 2 4 -3 + q + q - --- + --- + --- - --- + -- - -- - -- + -- + 5 q - 3 q + 16 14 12 10 8 6 4 2 q q q q q q q q 6 8 10 12 14 16 > q + 4 q - 4 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 209]][a, z]
Out[13]=	2 2 4 -2 4 6 2 z z 2 2 4 2 4 2 z 2 4 6 a - a + a - 2 z - -- + -- + 2 a z - 3 a z - z + ---- + 3 a z - z 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 209]][a, z]
Out[14]=	2 2 -2 4 6 2 z 3 5 2 2 z z 2 2 -a - a - a + --- + 6 a z + 3 a z - a z - 9 z + ---- - -- - 7 a z + a 4 2 a a 3 3 3 4 2 6 2 z 5 z 2 z 3 3 3 5 3 4 > 2 a z + 3 a z - -- + ---- + ---- - 10 a z - a z + 5 a z + 26 z - 5 3 a a a 4 4 5 5 5 5 z 7 z 2 4 4 4 6 4 z 11 z 8 z 5 > ---- + ---- + 20 a z + 3 a z - 3 a z + -- - ----- - ---- + 9 a z - 4 2 5 3 a a a a a 6 6 3 5 5 5 6 4 z 14 z 2 6 4 6 6 6 > 2 a z - 7 a z - 35 z + ---- - ----- - 27 a z - 9 a z + a z + 4 2 a a 7 7 8 8 z 2 z 7 3 7 5 7 8 10 z 2 8 > ---- - ---- - 18 a z - 5 a z + 3 a z + 15 z + ----- + 10 a z + 3 a 2 a a 9 4 8 7 z 9 3 9 10 2 10 > 5 a z + ---- + 12 a z + 5 a z + 2 z + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 209]], Vassiliev[3][Knot[11, Alternating, 209]]}
Out[15]=	{-3, 3}
In[16]:=	Kh[Knot[11, Alternating, 209]][q, t]
Out[16]=	11 1 2 1 6 2 8 6 11 -- + 13 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 12 11 3 3 2 5 2 5 3 > ----- + ---- + --- + 10 q t + 10 q t + 6 q t + 10 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a209

K11a208 K11a210