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

K11a225 K11a227

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X18,5,19,6 X20,7,21,8 X16,10,17,9 X14,12,15,11 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, -6, 8, -5, 9, -3, 10, -4, 11, -9}

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 16 14 2 10 22 8 6

Alexander Polynomial:

- 4t-2 + 18t-1 - 27 + 18t - 4t2

Conway Polynomial:

1 + 2z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a229, ...}

Determinant and Signature:

{71, -2}

Jones Polynomial:

- q-8 + 2q-7 - 4q-6 + 7q-5 - 9q-4 + 11q-3 - 11q-2 + 10q-1 - 7 + 5q - 3q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-26 - q-24 + q-22 - q-20 + 3q-16 - q-14 + q-12 + q-10 - q-8 + q-6 - 2q-4 + 2q-2 + 1 - q2 + 2q4 - q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + 1 - z4 - a2 - 2a2z2 - 2a2z4 + a4 + a4z2 - a4z4 + a6 + 2a6z2 - a8

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 + 6a-1z3 - 10a-1z5 + 3a-1z7 + 1 - 5z2 + 13z4 - 14z6 + 4z8 + 2az - 5az3 + 8az5 - 9az7 + 3az9 + a2 - 12a2z2 + 25a2z4 - 16a2z6 + 2a2z8 + a2z10 + 4a3z - 19a3z3 + 31a3z5 - 19a3z7 + 5a3z9 + a4 - 7a4z2 + 13a4z4 - 5a4z6 + a4z10 + 2a5z - 8a5z3 + 10a5z5 - 5a5z7 + 2a5z9 - a6 + 2a6z2 - a6z4 - 2a6z6 + 2a6z8 + 2a7z - 3a7z3 - 2a7z5 + 2a7z7 - a8 + 3a8z2 - 5a8z4 + 2a8z6 + 2a9z - 3a9z3 + a9z5

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=-2 is the signature of 11226. 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                 4 2     j = -1               6 3       j = -3             6 5         j = -5           5 5           j = -7         4 6             j = -9       3 5               j = -11     1 4                 j = -13   1 3                   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, 226]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 226]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[20, 7, 21, 8], > X[16, 10, 17, 9], X[14, 12, 15, 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, 226]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -6, 8, -5, 9, -3, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 226]]
Out[5]=	DTCode[4, 12, 18, 20, 16, 14, 2, 10, 22, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 226]][t]
Out[6]=	4 18 2 -27 - -- + -- + 18 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 226]][z]
Out[7]=	2 4 1 + 2 z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 226], Knot[11, Alternating, 229]}
In[9]:=	{KnotDet[Knot[11, Alternating, 226]], KnotSignature[Knot[11, Alternating, 226]]}
Out[9]=	{71, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 226]][q]
Out[10]=	-8 2 4 7 9 11 11 10 2 3 -7 - 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, 226]}
In[12]:=	A2Invariant[Knot[11, Alternating, 226]][q]
Out[12]=	-26 -24 -22 -20 3 -14 -12 -10 -8 -6 2 1 - q - q + q - q + --- - q + q + q - q + q - -- + 16 4 q q 2 2 4 6 8 10 > -- - q + 2 q - q - q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 226]][a, z]
Out[13]=	2 2 4 6 8 z 2 2 4 2 6 2 4 2 4 4 4 1 - a + a + a - a + -- - 2 a z + a z + 2 a z - z - 2 a z - a z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 226]][a, z]
Out[14]=	2 2 4 6 8 3 5 7 9 2 z 1 + a + a - a - a + 2 a z + 4 a z + 2 a z + 2 a z + 2 a z - 5 z + -- - 2 a 3 2 2 4 2 6 2 8 2 6 z 3 3 3 > 12 a z - 7 a z + 2 a z + 3 a z + ---- - 5 a z - 19 a z - a 4 5 3 7 3 9 3 4 3 z 2 4 4 4 6 4 > 8 a z - 3 a z - 3 a z + 13 z - ---- + 25 a z + 13 a z - a z - 2 a 5 8 4 10 z 5 3 5 5 5 7 5 9 5 6 > 5 a z - ----- + 8 a z + 31 a z + 10 a z - 2 a z + a z - 14 z + a 6 7 z 2 6 4 6 6 6 8 6 3 z 7 3 7 > -- - 16 a z - 5 a z - 2 a z + 2 a z + ---- - 9 a z - 19 a z - 2 a a 5 7 7 7 8 2 8 6 8 9 3 9 5 9 > 5 a z + 2 a z + 4 z + 2 a z + 2 a z + 3 a z + 5 a z + 2 a z + 2 10 4 10 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 226]], Vassiliev[3][Knot[11, Alternating, 226]]}
Out[15]=	{2, -5}
In[16]:=	Kh[Knot[11, Alternating, 226]][q, t]
Out[16]=	5 6 1 1 1 3 1 4 3 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 6 5 5 6 3 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 4 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 K11a226

K11a225 K11a227