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

K11a204 K11a206

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 18 20 14 2 10 22 8 6

Alexander Polynomial:

2t-3 - 10t-2 + 21t-1 - 25 + 21t - 10t2 + 2t3

Conway Polynomial:

1 - z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{91, -2}

Jones Polynomial:

q-7 - 3q-6 + 6q-5 - 10q-4 + 13q-3 - 14q-2 + 14q-1 - 12 + 9q - 5q2 + 3q3 - q4

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

{K11a86, ...}

A2 (sl(3)) Invariant:

q-22 - q-18 + 2q-16 - 2q-14 - q-12 + q-10 - 2q-8 + 3q-6 - q-4 + q-2 + 1 - 2q2 + 3q4 + q10 - q12

HOMFLY-PT Polynomial:

- 2a-2z2 - a-2z4 + 1 + 3z2 + 3z4 + z6 + a2 + a2z2 + 2a2z4 + a2z6 - 2a4 - 4a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial:

4a-3z3 - 4a-3z5 + a-3z7 - 6a-2z2 + 17a-2z4 - 13a-2z6 + 3a-2z8 + a-1z - 3a-1z3 + 9a-1z5 - 10a-1z7 + 3a-1z9 + 1 - 10z2 + 25z4 - 22z6 + 4z8 + z10 + 2az - 8az3 + 11az5 - 15az7 + 6az9 - a2 + a2z2 + 6a2z4 - 16a2z6 + 6a2z8 + a2z10 + a3z + 4a3z3 - 11a3z5 + 2a3z7 + 3a3z9 - 2a4 + 9a4z2 - 8a4z4 - 2a4z6 + 5a4z8 + a5z + 2a5z3 - 6a5z5 + 6a5z7 - a6 + 3a6z2 - 5a6z4 + 5a6z6 + a7z - 3a7z3 + 3a7z5 - a8z2 + a8z4

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

{-1, 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=-2 is the signature of 11205. 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 = 9                       1 j = 7                     2   j = 5                   3 1   j = 3                 6 2     j = 1               6 3       j = -1             8 6         j = -3           7 7           j = -5         6 7             j = -7       4 7               j = -9     2 6                 j = -11   1 4                   j = -13   2                     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]:=	Show[DrawMorseLink[Knot[11, Alternating, 205]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 205]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[18, 7, 19, 8], > X[20, 9, 21, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[10, 16, 11, 15], > X[22, 18, 1, 17], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 205]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -6, 8, -3, 9, -4, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 205]]
Out[5]=	DTCode[4, 12, 16, 18, 20, 14, 2, 10, 22, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 205]][t]
Out[6]=	2 10 21 2 3 -25 + -- - -- + -- + 21 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 205]][z]
Out[7]=	2 4 6 1 - z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 205]}
In[9]:=	{KnotDet[Knot[11, Alternating, 205]], KnotSignature[Knot[11, Alternating, 205]]}
Out[9]=	{91, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 205]][q]
Out[10]=	-7 3 6 10 13 14 14 2 3 4 -12 + q - -- + -- - -- + -- - -- + -- + 9 q - 5 q + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 86], Knot[11, Alternating, 205]}
In[12]:=	A2Invariant[Knot[11, Alternating, 205]][q]
Out[12]=	-22 -18 2 2 -12 -10 2 3 -4 -2 2 4 1 + q - q + --- - --- - q + q - -- + -- - q + q - 2 q + 3 q + 16 14 8 6 q q q q 10 12 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 205]][a, z]
Out[13]=	2 4 2 4 6 2 2 z 2 2 4 2 6 2 4 z 1 + a - 2 a + a + 3 z - ---- + a z - 4 a z + a z + 3 z - -- + 2 2 a a 2 4 4 4 6 2 6 > 2 a z - 2 a z + z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 205]][a, z]
Out[14]=	2 2 4 6 z 3 5 7 2 6 z 2 2 1 - a - 2 a - a + - + 2 a z + a z + a z + a z - 10 z - ---- + a z + a 2 a 3 3 4 2 6 2 8 2 4 z 3 z 3 3 3 5 3 > 9 a z + 3 a z - a z + ---- - ---- - 8 a z + 4 a z + 2 a z - 3 a a 4 5 7 3 4 17 z 2 4 4 4 6 4 8 4 4 z > 3 a z + 25 z + ----- + 6 a z - 8 a z - 5 a z + a z - ---- + 2 3 a a 5 6 9 z 5 3 5 5 5 7 5 6 13 z 2 6 > ---- + 11 a z - 11 a z - 6 a z + 3 a z - 22 z - ----- - 16 a z - a 2 a 7 7 4 6 6 6 z 10 z 7 3 7 5 7 8 > 2 a z + 5 a z + -- - ----- - 15 a z + 2 a z + 6 a z + 4 z + 3 a a 8 9 3 z 2 8 4 8 3 z 9 3 9 10 2 10 > ---- + 6 a z + 5 a z + ---- + 6 a z + 3 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 205]], Vassiliev[3][Knot[11, Alternating, 205]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 205]][q, t]
Out[16]=	7 8 1 2 1 4 2 6 4 7 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 6 7 7 6 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 3 q t + 6 q t + 2 q t + 3 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a205

K11a204 K11a206