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

K11a210 K11a212

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 20 18 14 2 22 10 8 6

Alexander Polynomial:

- 3t-2 + 17t-1 - 27 + 17t - 3t2

Conway Polynomial:

1 + 5z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{67, -2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 - q-26 + 2q-22 - 2q-20 + q-16 + 2q-12 + 2q-8 + q-6 - 2q-4 + 2q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 - a2z4 + 2a4 + 2a4z2 - a4z4 - a6 - a6z4 + a8 + 2a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 4az3 + 3az5 + a2z2 - 5a2z4 + 4a2z6 + 2a3z3 - 5a3z5 + 4a3z7 + 2a4 - 6a4z2 + 11a4z4 - 9a4z6 + 4a4z8 + a5z - 5a5z3 + 10a5z5 - 8a5z7 + 3a5z9 + a6 - 12a6z2 + 24a6z4 - 16a6z6 + 2a6z8 + a6z10 + 4a7z - 17a7z3 + 27a7z5 - 20a7z7 + 5a7z9 + a8 - 9a8z2 + 18a8z4 - 12a8z6 + a8z10 + a9z3 + 4a9z5 - 7a9z7 + 2a9z9 + a10 - 5a10z2 + 11a10z4 - 9a10z6 + 2a10z8 - 3a11z + 7a11z3 - 5a11z5 + a11z7

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

{5, -12}

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 11211. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     2   j = -1                   3 1   j = -3                 5 3     j = -5               5 2       j = -7             5 5         j = -9           5 5           j = -11         3 5             j = -13       3 5               j = -15     1 3                 j = -17   1 3                   j = -19   1                     j = -21 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, 211]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 211]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[20, 7, 21, 8], > X[18, 9, 19, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[22, 15, 1, 16], > X[10, 17, 11, 18], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 211]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -6, 8, -3, 9, -5, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 211]]
Out[5]=	DTCode[4, 12, 16, 20, 18, 14, 2, 22, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 211]][t]
Out[6]=	3 17 2 -27 - -- + -- + 17 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 211]][z]
Out[7]=	2 4 1 + 5 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 211]}
In[9]:=	{KnotDet[Knot[11, Alternating, 211]], KnotSignature[Knot[11, Alternating, 211]]}
Out[9]=	{67, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 211]][q]
Out[10]=	-10 2 4 6 8 10 10 10 7 5 -3 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q 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, 211]}
In[12]:=	A2Invariant[Knot[11, Alternating, 211]][q]
Out[12]=	-32 -30 -28 -26 2 2 -16 2 2 -6 2 2 -1 - q - q + q - q + --- - --- + q + --- + -- + q - -- + -- - 22 20 12 8 4 2 q q q q q q 2 4 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 211]][a, z]
Out[13]=	4 6 8 10 2 4 2 8 2 2 4 4 4 6 4 2 a - a + a - a + z + 2 a z + 2 a z - a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 211]][a, z]
Out[14]=	4 6 8 10 5 7 11 2 2 2 4 2 2 a + a + a + a + a z + 4 a z - 3 a z - z + a z - 6 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 12 a z - 9 a z - 5 a z - 4 a z + 2 a z - 5 a z - 17 a z + 9 3 11 3 4 2 4 4 4 6 4 8 4 > a z + 7 a z + z - 5 a z + 11 a z + 24 a z + 18 a z + 10 4 5 3 5 5 5 7 5 9 5 11 5 > 11 a z + 3 a z - 5 a z + 10 a z + 27 a z + 4 a z - 5 a z + 2 6 4 6 6 6 8 6 10 6 3 7 5 7 > 4 a z - 9 a z - 16 a z - 12 a z - 9 a z + 4 a z - 8 a z - 7 7 9 7 11 7 4 8 6 8 10 8 5 9 > 20 a z - 7 a z + a z + 4 a z + 2 a z + 2 a z + 3 a z + 7 9 9 9 6 10 8 10 > 5 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 211]], Vassiliev[3][Knot[11, Alternating, 211]]}
Out[15]=	{5, -12}
In[16]:=	Kh[Knot[11, Alternating, 211]][q, t]
Out[16]=	3 3 1 1 1 3 1 3 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 5 3 5 5 5 5 5 5 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 5 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a211

K11a210 K11a212