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

K11n20 K11n22

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X9,15,10,14 X11,19,12,18 X6,13,7,14 X15,20,16,21 X17,11,18,10 X19,22,20,1 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 -14 -18 6 -20 -10 -22 -16

Alexander Polynomial:

- t-3 + 5t-2 - 11t-1 + 15 - 11t + 5t2 - t3

Conway Polynomial:

1 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{927, K11n4, K11n172, ...}

Determinant and Signature:

{49, 0}

Jones Polynomial:

- q-3 + 3q-2 - 5q-1 + 8 - 8q + 8q2 - 7q3 + 5q4 - 3q5 + q6

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

{941, K11n4, ...}

A2 (sl(3)) Invariant:

- q-10 + q-6 - q-4 + 3q-2 + 1 + q2 + q4 - 2q6 + q8 - 2q10 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 3a-2 - 6a-2z2 - 4a-2z4 - a-2z6 + 4 + 5z2 + 2z4 - a2 - a2z2

Kauffman Polynomial:

2a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 8a-5z3 - 10a-5z5 + 3a-5z7 + a-4 - a-4z2 + 3a-4z4 - 8a-4z6 + 3a-4z8 - 5a-3z + 16a-3z3 - 17a-3z5 + 3a-3z7 + a-3z9 + 3a-2 - 10a-2z2 + 16a-2z4 - 15a-2z6 + 5a-2z8 - 5a-1z + 12a-1z3 - 8a-1z5 + a-1z7 + a-1z9 + 4 - 10z2 + 13z4 - 6z6 + 2z8 - 3az + 5az3 - az5 + az7 + a2 - 3a2z2 + 3a2z4 - a3z + a3z3

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

{0, -1}

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=0 is the signature of 1121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 2   j = 9               3 1   j = 7             4 2     j = 5           4 3       j = 3         4 4         j = 1       4 4           j = -1     2 5             j = -3   1 3               j = -5   2                 j = -7 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, NonAlternating, 21]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 21]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[9, 15, 10, 14], X[11, 19, 12, 18], X[6, 13, 7, 14], X[15, 20, 16, 21], > X[17, 11, 18, 10], X[19, 22, 20, 1], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 21]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 9, -6, -3, 7, 5, -8, 11, -9, 6, -10, > 8, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 21]]
Out[5]=	DTCode[4, 8, 12, 2, -14, -18, 6, -20, -10, -22, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 21]][t]
Out[6]=	-3 5 11 2 3 15 - t + -- - -- - 11 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 21]][z]
Out[7]=	4 6 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 27], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21], > Knot[11, NonAlternating, 172]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 21]], KnotSignature[Knot[11, NonAlternating, 21]]}
Out[9]=	{49, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 21]][q]
Out[10]=	-3 3 5 2 3 4 5 6 8 - q + -- - - - 8 q + 8 q - 7 q + 5 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 41], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 21]][q]
Out[12]=	-10 -6 -4 3 2 4 6 8 10 14 16 18 1 - q + q - q + -- + q + q - 2 q + q - 2 q + q - q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 21]][a, z]
Out[13]=	2 2 4 4 6 -4 3 2 2 2 z 6 z 2 2 4 z 4 z z 4 + a - -- - a + 5 z + ---- - ---- - a z + 2 z + -- - ---- - -- 2 4 2 4 2 2 a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 21]][a, z]
Out[14]=	2 2 -4 3 2 2 z 5 z 5 z 3 2 2 z z 4 + a + -- + a - --- - --- - --- - 3 a z - a z - 10 z + ---- - -- - 2 5 3 a 6 4 a a a a a 2 3 3 3 4 10 z 2 2 8 z 16 z 12 z 3 3 3 4 3 z > ----- - 3 a z + ---- + ----- + ----- + 5 a z + a z + 13 z - ---- + 2 5 3 a 6 a a a a 4 4 5 5 5 6 6 3 z 16 z 2 4 10 z 17 z 8 z 5 6 z 8 z > ---- + ----- + 3 a z - ----- - ----- - ---- - a z - 6 z + -- - ---- - 4 2 5 3 a 6 4 a a a a a a 6 7 7 7 8 8 9 9 15 z 3 z 3 z z 7 8 3 z 5 z z z > ----- + ---- + ---- + -- + a z + 2 z + ---- + ---- + -- + -- 2 5 3 a 4 2 3 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 21]], Vassiliev[3][Knot[11, NonAlternating, 21]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 21]][q, t]
Out[16]=	5 1 2 1 3 2 3 3 2 - + 4 q + ----- + ----- + ----- + ---- + --- + 4 q t + 4 q t + 4 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 4 q t + 3 q t + 4 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 K11n21

K11n20 K11n22