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

K11a218 K11a220

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 16 20 14 2 22 6 10 8

Alexander Polynomial:

- 4t-2 + 22t-1 - 35 + 22t - 4t2

Conway Polynomial:

1 + 6z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{87, -2}

Jones Polynomial:

- q-10 + 2q-9 - 5q-8 + 8q-7 - 11q-6 + 14q-5 - 13q-4 + 13q-3 - 10q-2 + 6q-1 - 3 + q

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 - 2q-26 - q-24 + 3q-22 - 2q-20 + q-18 + 2q-16 + 3q-12 - q-10 + 2q-8 - 3q-4 + 3q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2z2 - a2z4 + a4 - 2a4z4 + a6 + 2a6z2 - a6z4 + 2a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 3az3 + 3az5 + a2z2 - 5a2z4 + 5a2z6 + 9a3z3 - 12a3z5 + 7a3z7 + a4 - 9a4z2 + 19a4z4 - 16a4z6 + 7a4z8 + 2a5z - 5a5z7 + 4a5z9 - a6 - 13a6z2 + 30a6z4 - 27a6z6 + 7a6z8 + a6z10 + 6a7z - 17a7z3 + 20a7z5 - 18a7z7 + 6a7z9 - 5a8z2 + 14a8z4 - 14a8z6 + 2a8z8 + a8z10 + 3a9z3 - 5a9z7 + 2a9z9 + a10 - 3a10z2 + 9a10z4 - 8a10z6 + 2a10z8 - 4a11z + 8a11z3 - 5a11z5 + a11z7

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

{6, -15}

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 11219. 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                   4 1   j = -3                 7 3     j = -5               6 3       j = -7             7 7         j = -9           7 6           j = -11         4 7             j = -13       4 7               j = -15     1 4                 j = -17   1 4                   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, 219]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 219]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[16, 7, 17, 8], > X[20, 9, 21, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[22, 15, 1, 16], > X[6, 17, 7, 18], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 219]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -11, 5, -10, 6, -2, 7, -6, 8, -4, 9, -3, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 219]]
Out[5]=	DTCode[4, 12, 18, 16, 20, 14, 2, 22, 6, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 219]][t]
Out[6]=	4 22 2 -35 - -- + -- + 22 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 219]][z]
Out[7]=	2 4 1 + 6 z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 219]}
In[9]:=	{KnotDet[Knot[11, Alternating, 219]], KnotSignature[Knot[11, Alternating, 219]]}
Out[9]=	{87, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 219]][q]
Out[10]=	-10 2 5 8 11 14 13 13 10 6 -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, 219]}
In[12]:=	A2Invariant[Knot[11, Alternating, 219]][q]
Out[12]=	-32 -30 -28 2 -24 3 2 -18 2 3 -10 -1 - q - q + q - --- - q + --- - --- + q + --- + --- - q + 26 22 20 16 12 q q q q q 2 3 3 2 4 > -- - -- + -- - q + q 8 4 2 q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 219]][a, z]
Out[13]=	4 6 10 2 2 2 6 2 8 2 2 4 4 4 6 4 a + a - a + z + a z + 2 a z + 2 a z - a z - 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 219]][a, z]
Out[14]=	4 6 10 5 7 11 2 2 2 4 2 6 2 a - a + a + 2 a z + 6 a z - 4 a z - z + a z - 9 a z - 13 a z - 8 2 10 2 3 3 3 7 3 9 3 11 3 > 5 a z - 3 a z - 3 a z + 9 a z - 17 a z + 3 a z + 8 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > z - 5 a z + 19 a z + 30 a z + 14 a z + 9 a z + 3 a z - 3 5 7 5 11 5 2 6 4 6 6 6 8 6 > 12 a z + 20 a z - 5 a z + 5 a z - 16 a z - 27 a z - 14 a z - 10 6 3 7 5 7 7 7 9 7 11 7 4 8 > 8 a z + 7 a z - 5 a z - 18 a z - 5 a z + a z + 7 a z + 6 8 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 7 a z + 2 a z + 2 a z + 4 a z + 6 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 219]], Vassiliev[3][Knot[11, Alternating, 219]]}
Out[15]=	{6, -15}
In[16]:=	Kh[Knot[11, Alternating, 219]][q, t]
Out[16]=	3 4 1 1 1 4 1 4 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 4 7 7 6 7 7 6 3 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 7 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a219

K11a218 K11a220