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

K11a221 K11a223

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 14 16 2 10 22 8 6

Alexander Polynomial:

- 2t-3 + 11t-2 - 23t-1 + 29 - 23t + 11t2 - 2t3

Conway Polynomial:

1 + 3z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a204, ...}

Determinant and Signature:

{101, 0}

Jones Polynomial:

- q-7 + 2q-6 - 5q-5 + 9q-4 - 12q-3 + 16q-2 - 16q-1 + 15 - 12q + 8q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 - q-20 - 2q-16 + 2q-14 + 2q-12 - q-10 + 4q-8 - q-6 + q-4 + q-2 - 2 + 3q2 - 3q4 + q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 - z2 - 2z4 - z6 - a2z2 - 2a2z4 - a2z6 + 3a4 + 5a4z2 + 2a4z4 - 2a6 - a6z2

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 + 2a-2z2 - 8a-2z4 + 8a-2z6 + 5a-1z3 - 13a-1z5 + 10a-1z7 + z2 - z4 - 8z6 + 8z8 + 7az3 - 16az5 + 3az7 + 4az9 - 4a2z2 + 15a2z4 - 24a2z6 + 9a2z8 + a2z10 - a3z + 3a3z3 + 2a3z5 - 12a3z7 + 6a3z9 + 3a4 - 8a4z2 + 17a4z4 - 16a4z6 + 3a4z8 + a4z10 - 5a5z + 11a5z3 - 4a5z5 - 4a5z7 + 2a5z9 + 2a6 - 5a6z2 + 10a6z4 - 8a6z6 + 2a6z8 - 4a7z + 8a7z3 - 5a7z5 + a7z7

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

{3, -5}

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 11222. 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 = 9                       1 j = 7                     3   j = 5                   5 1   j = 3                 7 3     j = 1               8 5       j = -1             9 8         j = -3           7 7           j = -5         5 9             j = -7       4 7               j = -9     1 5                 j = -11   1 4                   j = -13   1                     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, 222]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 222]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[20, 7, 21, 8], > X[14, 10, 15, 9], X[16, 12, 17, 11], X[2, 13, 3, 14], X[10, 16, 11, 15], > X[22, 17, 1, 18], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 222]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -5, 8, -6, 9, -3, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 222]]
Out[5]=	DTCode[4, 12, 18, 20, 14, 16, 2, 10, 22, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 222]][t]
Out[6]=	2 11 23 2 3 29 - -- + -- - -- - 23 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 222]][z]
Out[7]=	2 4 6 1 + 3 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 204], Knot[11, Alternating, 222]}
In[9]:=	{KnotDet[Knot[11, Alternating, 222]], KnotSignature[Knot[11, Alternating, 222]]}
Out[9]=	{101, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 222]][q]
Out[10]=	-7 2 5 9 12 16 16 2 3 4 15 - q + -- - -- + -- - -- + -- - -- - 12 q + 8 q - 4 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, 222]}
In[12]:=	A2Invariant[Knot[11, Alternating, 222]][q]
Out[12]=	-22 -20 2 2 2 -10 4 -6 -4 -2 2 -2 - q - q - --- + --- + --- - q + -- - q + q + q + 3 q - 16 14 12 8 q q q q 4 6 8 10 12 > 3 q + q + q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 222]][a, z]
Out[13]=	2 4 4 6 2 z 2 2 4 2 6 2 4 z 2 4 3 a - 2 a - z + -- - a z + 5 a z - a z - 2 z + -- - 2 a z + 2 2 a a 4 4 6 2 6 > 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 222]][a, z]
Out[14]=	2 4 6 3 5 7 2 2 z 2 2 4 2 3 a + 2 a - a z - 5 a z - 4 a z + z + ---- - 4 a z - 8 a z - 2 a 3 3 4 6 2 2 z 5 z 3 3 3 5 3 7 3 4 z > 5 a z - ---- + ---- + 7 a z + 3 a z + 11 a z + 8 a z - z + -- - 3 a 4 a a 4 5 5 8 z 2 4 4 4 6 4 4 z 13 z 5 3 5 > ---- + 15 a z + 17 a z + 10 a z + ---- - ----- - 16 a z + 2 a z - 2 3 a a a 6 7 5 5 7 5 6 8 z 2 6 4 6 6 6 10 z > 4 a z - 5 a z - 8 z + ---- - 24 a z - 16 a z - 8 a z + ----- + 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 6 8 > 3 a z - 12 a z - 4 a z + a z + 8 z + 9 a z + 3 a z + 2 a z + 9 3 9 5 9 2 10 4 10 > 4 a z + 6 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 222]], Vassiliev[3][Knot[11, Alternating, 222]]}
Out[15]=	{3, -5}
In[16]:=	Kh[Knot[11, Alternating, 222]][q, t]
Out[16]=	8 1 1 1 4 1 5 4 7 - + 8 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 5 9 7 7 9 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 5 q t + 7 q t + 3 q t + 5 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a222

K11a221 K11a223