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

K11a222 K11a224

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 16 2 22 8 10 6 14

Alexander Polynomial:

- t-4 + 5t-3 - 10t-2 + 14t-1 - 15 + 14t - 10t2 + 5t3 - t4

Conway Polynomial:

1 + 3z2 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11n148, ...}

Determinant and Signature:

{75, 6}

Jones Polynomial:

q - 2q2 + 5q3 - 7q4 + 10q5 - 11q6 + 11q7 - 11q8 + 8q9 - 5q10 + 3q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q4 + 2q8 + q10 + 2q14 - 2q16 + 2q18 - 2q20 - q22 - 2q26 + 2q28 + q32 - q36

HOMFLY-PT Polynomial:

- a-10 - 3a-10z2 - a-10z4 + 3a-8 + 11a-8z2 + 9a-8z4 + 2a-8z6 - 5a-6 - 13a-6z2 - 13a-6z4 - 6a-6z6 - a-6z8 + 4a-4 + 8a-4z2 + 5a-4z4 + a-4z6

Kauffman Polynomial:

a-15z3 - a-14z2 + 3a-14z4 - 3a-13z3 + 5a-13z5 - 8a-12z4 + 7a-12z6 - 3a-11z + 12a-11z3 - 20a-11z5 + 9a-11z7 + a-10 - 6a-10z2 + 14a-10z4 - 21a-10z6 + 8a-10z8 - 6a-9z + 30a-9z3 - 25a-9z5 - 3a-9z7 + 4a-9z9 + 3a-8 - 22a-8z2 + 55a-8z4 - 44a-8z6 + 8a-8z8 + a-8z10 - 6a-7z + 9a-7z3 + 15a-7z5 - 22a-7z7 + 6a-7z9 + 5a-6 - 27a-6z2 + 43a-6z4 - 22a-6z6 + a-6z8 + a-6z10 - 3a-5z - 5a-5z3 + 15a-5z5 - 10a-5z7 + 2a-5z9 + 4a-4 - 12a-4z2 + 13a-4z4 - 6a-4z6 + a-4z8

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

{3, 6}

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=6 is the signature of 11223. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 25                       1 j = 23                     2   j = 21                   3 1   j = 19                 5 2     j = 17               6 3       j = 15             5 5         j = 13           6 6           j = 11         4 5             j = 9       3 6               j = 7     2 4                 j = 5   1 4                   j = 3   1                     j = 1 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, 223]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 223]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[20, 8, 21, 7], > X[16, 10, 17, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[8, 16, 9, 15], > X[10, 18, 11, 17], X[6, 20, 7, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 223]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 223]]
Out[5]=	DTCode[4, 12, 18, 20, 16, 2, 22, 8, 10, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 223]][t]
Out[6]=	-4 5 10 14 2 3 4 -15 - t + -- - -- + -- + 14 t - 10 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 223]][z]
Out[7]=	2 6 8 1 + 3 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 223], Knot[11, NonAlternating, 148]}
In[9]:=	{KnotDet[Knot[11, Alternating, 223]], KnotSignature[Knot[11, Alternating, 223]]}
Out[9]=	{75, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 223]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 5 q - 7 q + 10 q - 11 q + 11 q - 11 q + 8 q - 5 q + 3 q - 12 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 223]}
In[12]:=	A2Invariant[Knot[11, Alternating, 223]][q]
Out[12]=	4 8 10 14 16 18 20 22 26 28 32 q + 2 q + q + 2 q - 2 q + 2 q - 2 q - q - 2 q + 2 q + q - 36 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 223]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 3 5 4 3 z 11 z 13 z 8 z z 9 z 13 z -a + -- - -- + -- - ---- + ----- - ----- + ---- - --- + ---- - ----- + 8 6 4 10 8 6 4 10 8 6 a a a a a a a a a a 4 6 6 6 8 5 z 2 z 6 z z z > ---- + ---- - ---- + -- - -- 4 8 6 4 6 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 223]][a, z]
Out[14]=	2 2 2 2 -10 3 5 4 3 z 6 z 6 z 3 z z 6 z 22 z 27 z a + -- + -- + -- - --- - --- - --- - --- - --- - ---- - ----- - ----- - 8 6 4 11 9 7 5 14 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 12 z z 3 z 12 z 30 z 9 z 5 z 3 z 8 z 14 z > ----- + --- - ---- + ----- + ----- + ---- - ---- + ---- - ---- + ----- + 4 15 13 11 9 7 5 14 12 10 a a a a a a a a a a 4 4 4 5 5 5 5 5 6 55 z 43 z 13 z 5 z 20 z 25 z 15 z 15 z 7 z > ----- + ----- + ----- + ---- - ----- - ----- + ----- + ----- + ---- - 8 6 4 13 11 9 7 5 12 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 21 z 44 z 22 z 6 z 9 z 3 z 22 z 10 z 8 z 8 z > ----- - ----- - ----- - ---- + ---- - ---- - ----- - ----- + ---- + ---- + 10 8 6 4 11 9 7 5 10 8 a a a a a a a a a a 8 8 9 9 9 10 10 z z 4 z 6 z 2 z z z > -- + -- + ---- + ---- + ---- + --- + --- 6 4 9 7 5 8 6 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 223]], Vassiliev[3][Knot[11, Alternating, 223]]}
Out[15]=	{3, 6}
In[16]:=	Kh[Knot[11, Alternating, 223]][q, t]
Out[16]=	3 5 5 7 q q q 7 9 9 2 11 2 11 3 4 q + 2 q + -- + -- + -- + 4 q t + 3 q t + 6 q t + 4 q t + 5 q t + 2 t t t 13 3 13 4 15 4 15 5 17 5 17 6 > 6 q t + 6 q t + 5 q t + 5 q t + 6 q t + 3 q t + 19 6 19 7 21 7 21 8 23 8 25 9 > 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a223

K11a222 K11a224