K11a73

K11a72

K11a74

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	K11a73 Quick Notes

K11a73 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X22,17,1,18 X6,20,7,19 X8,21,9,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -7, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 12 14 16 2 18 20 22 6 8
Conway Notation	[8*211]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a73/ThurstonBennequinNumber Hyperbolic Volume 18.4184 A-Polynomial See Data:K11a73/A-polynomial

[edit Notes for K11a73's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a73's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+21t^{2}-37t+45-37t^{-1}+21t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 177, 0 }
Jones polynomial	$-q^{5}+5q^{4}-11q^{3}+18q^{2}-25q+29-28q^{-1}+25q^{-2}-18q^{-3}+11q^{-4}-5q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+4z^{6}+a^{4}z^{4}-5a^{2}z^{4}-2z^{4}a^{-2}+5z^{4}+a^{4}z^{2}-a^{2}z^{2}-a^{4}+3a^{2}+a^{-2}-2$
Kauffman polynomial (db, data sources)	$3a^{2}z^{10}+3z^{10}+9a^{3}z^{9}+19az^{9}+10z^{9}a^{-1}+10a^{4}z^{8}+21a^{2}z^{8}+14z^{8}a^{-2}+25z^{8}+5a^{5}z^{7}-8a^{3}z^{7}-23az^{7}+z^{7}a^{-1}+11z^{7}a^{-3}+a^{6}z^{6}-21a^{4}z^{6}-60a^{2}z^{6}-18z^{6}a^{-2}+5z^{6}a^{-4}-61z^{6}-9a^{5}z^{5}-14a^{3}z^{5}-16az^{5}-26z^{5}a^{-1}-14z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+13a^{4}z^{4}+42a^{2}z^{4}+6z^{4}a^{-2}-4z^{4}a^{-4}+38z^{4}+4a^{5}z^{3}+14a^{3}z^{3}+23az^{3}+18z^{3}a^{-1}+5z^{3}a^{-3}-2a^{4}z^{2}-5a^{2}z^{2}-3z^{2}-2a^{3}z-4az-2za^{-1}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$q^{18}-2q^{16}-q^{14}+2q^{12}-4q^{10}+6q^{8}+4q^{2}-6+5q^{-2}-5q^{-4}+q^{-6}+3q^{-8}-3q^{-10}+3q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-4q^{92}+11q^{90}-24q^{88}+36q^{86}-43q^{84}+30q^{82}+20q^{80}-101q^{78}+212q^{76}-304q^{74}+312q^{72}-190q^{70}-98q^{68}+491q^{66}-854q^{64}+1023q^{62}-847q^{60}+283q^{58}+525q^{56}-1305q^{54}+1734q^{52}-1579q^{50}+834q^{48}+250q^{46}-1270q^{44}+1787q^{42}-1587q^{40}+759q^{38}+346q^{36}-1218q^{34}+1474q^{32}-989q^{30}-18q^{28}+1115q^{26}-1794q^{24}+1733q^{22}-901q^{20}-408q^{18}+1698q^{16}-2465q^{14}+2411q^{12}-1509q^{10}+86q^{8}+1352q^{6}-2298q^{4}+2402q^{2}-1665+400q^{-2}+860q^{-4}-1617q^{-6}+1594q^{-8}-856q^{-10}-216q^{-12}+1119q^{-14}-1449q^{-16}+1056q^{-18}-149q^{-20}-885q^{-22}+1595q^{-24}-1680q^{-26}+1162q^{-28}-252q^{-30}-678q^{-32}+1298q^{-34}-1452q^{-36}+1156q^{-38}-583q^{-40}-34q^{-42}+503q^{-44}-720q^{-46}+692q^{-48}-493q^{-50}+241q^{-52}-8q^{-54}-149q^{-56}+206q^{-58}-199q^{-60}+140q^{-62}-72q^{-64}+21q^{-66}+16q^{-68}-28q^{-70}+27q^{-72}-20q^{-74}+10q^{-76}-4q^{-78}+q^{-80}$

Further Quantum Invariants

K11a73 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a73"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{4}-7t^{3}+21t^{2}-37t+45-37t^{-1}+21t^{-2}-7t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+z^{6}-z^{4}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 177, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{5}+5q^{4}-11q^{3}+18q^{2}-25q+29-28q^{-1}+25q^{-2}-18q^{-3}+11q^{-4}-5q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+4z^{6}+a^{4}z^{4}-5a^{2}z^{4}-2z^{4}a^{-2}+5z^{4}+a^{4}z^{2}-a^{2}z^{2}-a^{4}+3a^{2}+a^{-2}-2$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$3a^{2}z^{10}+3z^{10}+9a^{3}z^{9}+19az^{9}+10z^{9}a^{-1}+10a^{4}z^{8}+21a^{2}z^{8}+14z^{8}a^{-2}+25z^{8}+5a^{5}z^{7}-8a^{3}z^{7}-23az^{7}+z^{7}a^{-1}+11z^{7}a^{-3}+a^{6}z^{6}-21a^{4}z^{6}-60a^{2}z^{6}-18z^{6}a^{-2}+5z^{6}a^{-4}-61z^{6}-9a^{5}z^{5}-14a^{3}z^{5}-16az^{5}-26z^{5}a^{-1}-14z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+13a^{4}z^{4}+42a^{2}z^{4}+6z^{4}a^{-2}-4z^{4}a^{-4}+38z^{4}+4a^{5}z^{3}+14a^{3}z^{3}+23az^{3}+18z^{3}a^{-1}+5z^{3}a^{-3}-2a^{4}z^{2}-5a^{2}z^{2}-3z^{2}-2a^{3}z-4az-2za^{-1}-a^{4}-3a^{2}-a^{-2}-2$

Vassiliev invariants

V2 and V3:

(0, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $-8$ $0$ $16$ $8$ $0$ $-{\frac {80}{3}}$ $-{\frac {32}{3}}$ $-8$ $0$ $32$ $0$ $0$ $88$ $-56$ ${\frac {328}{3}}$ $-{\frac {8}{3}}$ $24$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 K11a73. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   7 1   -6 5                 11 4     7 3               14 7       -7 1             15 11         4 -1           14 15           1 -3         11 14             -3 -5       7 14               7 -7     4 11                 -7 -9   1 7                   6 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[11, Alternating, 73]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 73]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], X[16, 10, 17, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[20, 16, 21, 15], X[22, 17, 1, 18], X[6, 20, 7, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 73]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -7, 10, -8, 11, -9]
In[5]:=	BR[Knot[11, Alternating, 73]]
Out[5]=	BR[Knot[11, Alternating, 73]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 73]][t]
Out[6]=	-4 7 21 37 2 3 4 45 + t - -- + -- - -- - 37 t + 21 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 73]][z]
Out[7]=	4 6 8 1 - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 73]}
In[9]:=	{KnotDet[Knot[11, Alternating, 73]], KnotSignature[Knot[11, Alternating, 73]]}
Out[9]=	{177, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 73]][q]
Out[10]=	-6 5 11 18 25 28 2 3 4 5 29 + q - -- + -- - -- + -- - -- - 25 q + 18 q - 11 q + 5 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 73]}
In[12]:=	A2Invariant[Knot[11, Alternating, 73]][q]
Out[12]=	-18 2 -14 2 4 6 4 2 4 6 -6 + q - --- - q + --- - --- + -- + -- + 5 q - 5 q + q + 16 12 10 8 2 q q q q q 8 10 12 14 3 q - 3 q + 3 q - q
In[13]:=	Kauffman[Knot[11, Alternating, 73]][a, z]
Out[13]=	-2 2 4 2 z 3 2 2 2 -2 - a - 3 a - a - --- - 4 a z - 2 a z - 3 z - 5 a z - a 3 3 4 2 5 z 18 z 3 3 3 5 3 4 2 a z + ---- + ----- + 23 a z + 14 a z + 4 a z + 38 z - 3 a a 4 4 5 5 5 4 z 6 z 2 4 4 4 6 4 z 14 z 26 z ---- + ---- + 42 a z + 13 a z - a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 5 z 18 z 2 6 16 a z - 14 a z - 9 a z - 61 z + ---- - ----- - 60 a z - 4 2 a a 7 7 4 6 6 6 11 z z 7 3 7 5 7 8 21 a z + a z + ----- + -- - 23 a z - 8 a z + 5 a z + 25 z + 3 a a 8 9 14 z 2 8 4 8 10 z 9 3 9 10 ----- + 21 a z + 10 a z + ----- + 19 a z + 9 a z + 3 z + 2 a a 2 10 3 a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 73]], Vassiliev[3][Knot[11, Alternating, 73]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[11, Alternating, 73]][q, t]
Out[15]=	15 1 4 1 7 4 11 7 -- + 15 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 14 11 14 14 3 3 2 5 2 ----- + ----- + ---- + --- + 11 q t + 14 q t + 7 q t + 11 q t + 5 2 3 2 3 q t q t q t q t 5 3 7 3 7 4 9 4 11 5 4 q t + 7 q t + q t + 4 q t + q t