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

K11a223 K11a225

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 X10,16,11,15 X8,18,9,17 X6,20,7,19 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 16 2 22 10 8 6 14

Alexander Polynomial:

- 2t-3 + 10t-2 - 20t-1 + 25 - 20t + 10t2 - 2t3

Conway Polynomial:

1 + 2z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1092, K11a153, K11n35, K11n43, ...}

Determinant and Signature:

{89, 4}

Jones Polynomial:

1 - 2q + 5q2 - 8q3 + 12q4 - 14q5 + 14q6 - 13q7 + 10q8 - 6q9 + 3q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 + q4 + 2q6 - 2q8 + 3q10 - q12 - q14 + q16 - 3q18 + 2q20 - q22 + q24 + 2q26 - 2q28 + q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 2a-8 + 4a-8z2 + 2a-8z4 - a-6 - a-6z2 - 2a-6z4 - a-6z6 - a-4 - 3a-4z2 - 3a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

- 2a-13z3 + a-13z5 + 2a-12z2 - 6a-12z4 + 3a-12z6 - 2a-11z + 8a-11z3 - 11a-11z5 + 5a-11z7 + a-10 - 3a-10z2 + 9a-10z4 - 10a-10z6 + 5a-10z8 - a-9z + 5a-9z3 - 2a-9z5 - 2a-9z7 + 3a-9z9 + 2a-8 - 10a-8z2 + 16a-8z4 - 11a-8z6 + 4a-8z8 + a-8z10 + 3a-7z - 17a-7z3 + 20a-7z5 - 12a-7z7 + 5a-7z9 + a-6 - 3a-6z2 - a-6z4 - a-6z6 + a-6z8 + a-6z10 + 3a-5z - 8a-5z3 + 4a-5z5 - 3a-5z7 + 2a-5z9 - a-4 + 7a-4z2 - 6a-4z4 - 2a-4z6 + 2a-4z8 + a-3z + 4a-3z3 - 6a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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

{2, 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=4 is the signature of 11224. 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 = 23                       1 j = 21                     2   j = 19                   4 1   j = 17                 6 2     j = 15               7 4       j = 13             7 6         j = 11           7 7           j = 9         5 7             j = 7       3 7               j = 5     2 5                 j = 3   1 4                   j = 1   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, 224]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 224]]
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[10, 16, 11, 15], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 224]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -8, 6, -2, 7, -11, 8, -5, 9, -3, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 224]]
Out[5]=	DTCode[4, 12, 18, 20, 16, 2, 22, 10, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 224]][t]
Out[6]=	2 10 20 2 3 25 - -- + -- - -- - 20 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 224]][z]
Out[7]=	2 4 6 1 + 2 z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 92], Knot[11, Alternating, 153], Knot[11, Alternating, 224], > Knot[11, NonAlternating, 35], Knot[11, NonAlternating, 43]}
In[9]:=	{KnotDet[Knot[11, Alternating, 224]], KnotSignature[Knot[11, Alternating, 224]]}
Out[9]=	{89, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 224]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 8 q + 12 q - 14 q + 14 q - 13 q + 10 q - 6 q + 3 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 224]}
In[12]:=	A2Invariant[Knot[11, Alternating, 224]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 24 1 + q + 2 q - 2 q + 3 q - q - q + q - 3 q + 2 q - q + q + 26 28 30 34 > 2 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 224]][a, z]
Out[13]=	2 2 2 2 2 4 4 -10 2 -6 -4 2 z 4 z z 3 z 3 z 2 z 2 z -a + -- - a - a + -- - --- + ---- - -- - ---- + ---- + ---- - ---- - 8 2 10 8 6 4 2 8 6 a a a a a a a a a 4 4 6 6 3 z z z z > ---- + -- - -- - -- 4 2 6 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 224]][a, z]
Out[14]=	2 2 2 -10 2 -6 -4 2 2 z z 3 z 3 z z 2 z 3 z 10 z a + -- + a - a - -- - --- - -- + --- + --- + -- + ---- - ---- - ----- - 8 2 11 9 7 5 3 12 10 8 a a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 3 z 7 z 5 z 2 z 8 z 5 z 17 z 8 z 4 z 6 z > ---- + ---- + ---- - ---- + ---- + ---- - ----- - ---- + ---- - ---- + 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 9 z 16 z z 6 z 4 z z 11 z 2 z 20 z 4 z > ---- + ----- - -- - ---- - ---- + --- - ----- - ---- + ----- + ---- - 10 8 6 4 2 13 11 9 7 5 a a a a a a a a a a 5 6 6 6 6 6 6 7 7 7 7 6 z 3 z 10 z 11 z z 2 z z 5 z 2 z 12 z 3 z > ---- + ---- - ----- - ----- - -- - ---- + -- + ---- - ---- - ----- - ---- + 3 12 10 8 6 4 2 11 9 7 5 a a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 2 z 5 z 4 z z 2 z 3 z 5 z 2 z z z > ---- + ---- + ---- + -- + ---- + ---- + ---- + ---- + --- + --- 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 224]], Vassiliev[3][Knot[11, Alternating, 224]]}
Out[15]=	{2, 5}
In[16]:=	Kh[Knot[11, Alternating, 224]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 5 q t + 3 q t + 7 q t + 5 q t + 7 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 7 q t + 7 q t + 7 q t + 6 q t + 7 q t + 4 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a224

K11a223 K11a225