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

K11a220 K11a222

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 X6,19,7,20 X8,21,9,22

Gauss Code:

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

Alexander Polynomial:

- t-4 + 5t-3 - 10t-2 + 15t-1 - 17 + 15t - 10t2 + 5t3 - t4

Conway Polynomial:

1 + 4z2 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a259, ...}

Determinant and Signature:

{79, -2}

Jones Polynomial:

- q-8 + 2q-7 - 5q-6 + 8q-5 - 10q-4 + 13q-3 - 12q-2 + 11q-1 - 8 + 5q - 3q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-20 - 2q-18 + 2q-16 + 3q-12 + 3q-10 + 3q-6 - 3q-4 + q-2 - 1 - q2 + q4 - q6 + q8

HOMFLY-PT Polynomial:

1 + 4z2 + 4z4 + z6 - 4a2 - 12a2z2 - 13a2z4 - 6a2z6 - a2z8 + 8a4 + 16a4z2 + 10a4z4 + 2a4z6 - 4a6 - 4a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 - a-1z + 7a-1z3 - 10a-1z5 + 3a-1z7 + 1 - 4z2 + 11z4 - 13z6 + 4z8 - 3az + 5az3 + az5 - 7az7 + 3az9 + 4a2 - 15a2z2 + 25a2z4 - 17a2z6 + 3a2z8 + a2z10 - 5a3z + 2a3z3 + 14a3z5 - 14a3z7 + 5a3z9 + 8a4 - 21a4z2 + 24a4z4 - 11a4z6 + 2a4z8 + a4z10 - 7a5z + 11a5z3 - 4a5z5 - a5z7 + 2a5z9 + 4a6 - 10a6z2 + 9a6z4 - 6a6z6 + 3a6z8 - 2a7z + 4a7z3 - 6a7z5 + 3a7z7 + a8z2 - 4a8z4 + 2a8z6 + 2a9z - 3a9z3 + a9z5

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

{4, -8}

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 11221. 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                     2   j = 3                   3 1   j = 1                 5 2     j = -1               6 3       j = -3             7 6         j = -5           6 5           j = -7         4 7             j = -9       4 6               j = -11     1 4                 j = -13   1 4                   j = -15   1                     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, 221]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 221]]
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[6, 19, 7, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 221]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -11, 5, -8, 6, -2, 7, -5, 8, -6, 9, -3, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 221]]
Out[5]=	DTCode[4, 12, 18, 20, 14, 16, 2, 10, 22, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 221]][t]
Out[6]=	-4 5 10 15 2 3 4 -17 - t + -- - -- + -- + 15 t - 10 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 221]][z]
Out[7]=	2 6 8 1 + 4 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 221], Knot[11, Alternating, 259]}
In[9]:=	{KnotDet[Knot[11, Alternating, 221]], KnotSignature[Knot[11, Alternating, 221]]}
Out[9]=	{79, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 221]][q]
Out[10]=	-8 2 5 8 10 13 12 11 2 3 -8 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - 3 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, 221]}
In[12]:=	A2Invariant[Knot[11, Alternating, 221]][q]
Out[12]=	-24 -22 -20 2 2 3 3 3 3 -2 2 4 -1 - q - q - q - --- + --- + --- + --- + -- - -- + q - q + q - 18 16 12 10 6 4 q q q q q q 6 8 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 221]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 1 - 4 a + 8 a - 4 a + 4 z - 12 a z + 16 a z - 4 a z + 4 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 13 a z + 10 a z - a z + z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 221]][a, z]
Out[14]=	2 4 6 z 3 5 7 9 2 1 + 4 a + 8 a + 4 a - - - 3 a z - 5 a z - 7 a z - 2 a z + 2 a z - 4 z + a 2 3 z 2 2 4 2 6 2 8 2 7 z 3 3 3 > -- - 15 a z - 21 a z - 10 a z + a z + ---- + 5 a z + 2 a z + 2 a a 4 5 3 7 3 9 3 4 3 z 2 4 4 4 > 11 a z + 4 a z - 3 a z + 11 z - ---- + 25 a z + 24 a z + 2 a 5 6 4 8 4 10 z 5 3 5 5 5 7 5 9 5 > 9 a z - 4 a z - ----- + a z + 14 a z - 4 a z - 6 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 3 z 7 > 13 z + -- - 17 a z - 11 a z - 6 a z + 2 a z + ---- - 7 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 9 > 14 a z - a z + 3 a z + 4 z + 3 a z + 2 a z + 3 a z + 3 a z + 3 9 5 9 2 10 4 10 > 5 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 221]], Vassiliev[3][Knot[11, Alternating, 221]]}
Out[15]=	{4, -8}
In[16]:=	Kh[Knot[11, Alternating, 221]][q, t]
Out[16]=	6 6 1 1 1 4 1 4 4 6 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 4 7 6 5 7 3 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 5 q t + 2 q t + 3 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 + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a221

K11a220 K11a222