9 44

9_43

9_45

Visit 9 44's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 44's page at Knotilus! Visit 9 44's page at the original Knot Atlas! 9 44 Quick Notes

9 44 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13
Gauss code	-1, 4, -3, 1, -2, -9, 5, 3, -4, 2, 7, -8, 9, -5, 6, -7, 8, -6
Dowker-Thistlethwaite code	4 8 10 -14 2 -16 -6 -18 -12
Conway Notation	[22,21,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-3] Hyperbolic Volume 7.40677 A-Polynomial See Data:9 44/A-polynomial

[edit Notes for 9 44's three dimensional invariants] 9_44 has girth 4. See arXiv:math.GT/0508590 and a forthcoming paper by the same authors.

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 9 44's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-4t+7-4t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 17, 0 }
Jones polynomial	$q^{2}-2q+3-3q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+3a^{2}-2z^{2}-2+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$-q^{16}+2q^{8}+q^{6}+q^{4}-1-q^{-4}+q^{-6}+q^{-8}$
The G2 invariant	$q^{80}-q^{78}+2q^{76}-3q^{74}+q^{72}-4q^{68}+6q^{66}-5q^{64}+3q^{62}-q^{60}-4q^{58}+5q^{56}-4q^{54}+2q^{50}-4q^{48}+4q^{46}-4q^{42}+7q^{40}-5q^{38}+3q^{36}-2q^{32}+6q^{30}-4q^{28}+6q^{26}-2q^{24}+2q^{22}+4q^{20}-4q^{18}+4q^{16}-2q^{14}+q^{12}+3q^{10}-4q^{8}+2q^{6}-4q^{2}+5-6q^{-2}+2q^{-6}-6q^{-8}+5q^{-10}-3q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-32}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 9_44.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}+q^{5}+q^{-1}-q^{-3}+q^{-5}$
2	$q^{32}-q^{30}-2q^{28}+2q^{26}+q^{24}-2q^{22}+2q^{18}-q^{16}-q^{14}+2q^{12}-q^{8}+q^{6}+q^{4}-q^{2}-1+3q^{-2}-2q^{-6}+2q^{-8}+q^{-10}-q^{-12}$
3	$-q^{63}+q^{61}+2q^{59}-3q^{55}-3q^{53}+3q^{51}+4q^{49}-4q^{45}-3q^{43}+3q^{41}+5q^{39}-q^{37}-6q^{35}-3q^{33}+6q^{31}+4q^{29}-4q^{27}-4q^{25}+4q^{23}+5q^{21}-3q^{19}-4q^{17}+2q^{15}+3q^{13}-2q^{11}-3q^{9}+3q^{5}+3q^{3}-2q-4q^{-1}+2q^{-3}+7q^{-5}+3q^{-7}-7q^{-9}-4q^{-11}+5q^{-13}+4q^{-15}-2q^{-17}-5q^{-19}+3q^{-23}+q^{-25}-q^{-29}$
4	$q^{104}-q^{102}-2q^{100}+q^{96}+5q^{94}+q^{92}-3q^{90}-5q^{88}-6q^{86}+5q^{84}+7q^{82}+5q^{80}-10q^{76}-7q^{74}-q^{72}+9q^{70}+14q^{68}+q^{66}-11q^{64}-16q^{62}-5q^{60}+15q^{58}+17q^{56}+2q^{54}-18q^{52}-18q^{50}+5q^{48}+21q^{46}+14q^{44}-9q^{42}-20q^{40}-5q^{38}+14q^{36}+13q^{34}-4q^{32}-15q^{30}-5q^{28}+9q^{26}+8q^{24}-2q^{22}-9q^{20}-3q^{18}+7q^{16}+6q^{14}+2q^{12}-4q^{10}-7q^{8}-q^{6}+6q^{4}+13q^{2}+3-15q^{-2}-15q^{-4}-q^{-6}+22q^{-8}+20q^{-10}-9q^{-12}-25q^{-14}-15q^{-16}+15q^{-18}+26q^{-20}+5q^{-22}-14q^{-24}-18q^{-26}-2q^{-28}+13q^{-30}+9q^{-32}-7q^{-36}-5q^{-38}+q^{-40}+2q^{-42}+2q^{-44}-q^{-48}$
5	$-q^{155}+q^{153}+2q^{151}-q^{147}-3q^{145}-3q^{143}-q^{141}+5q^{139}+7q^{137}+4q^{135}-q^{133}-7q^{131}-11q^{129}-8q^{127}+5q^{125}+11q^{123}+13q^{121}+9q^{119}-4q^{117}-16q^{115}-20q^{113}-10q^{111}+6q^{109}+24q^{107}+29q^{105}+13q^{103}-16q^{101}-37q^{99}-34q^{97}-8q^{95}+32q^{93}+50q^{91}+32q^{89}-13q^{87}-54q^{85}-54q^{83}-12q^{81}+43q^{79}+67q^{77}+38q^{75}-25q^{73}-67q^{71}-53q^{69}+3q^{67}+59q^{65}+62q^{63}+13q^{61}-46q^{59}-62q^{57}-22q^{55}+32q^{53}+54q^{51}+25q^{49}-23q^{47}-45q^{45}-23q^{43}+18q^{41}+34q^{39}+17q^{37}-13q^{35}-26q^{33}-10q^{31}+12q^{29}+19q^{27}+8q^{25}-9q^{23}-16q^{21}-10q^{19}+4q^{17}+17q^{15}+17q^{13}+6q^{11}-14q^{9}-29q^{7}-21q^{5}+9q^{3}+41q+40q^{-1}+5q^{-3}-44q^{-5}-63q^{-7}-27q^{-9}+42q^{-11}+81q^{-13}+49q^{-15}-23q^{-17}-82q^{-19}-70q^{-21}+q^{-23}+69q^{-25}+78q^{-27}+19q^{-29}-44q^{-31}-66q^{-33}-36q^{-35}+18q^{-37}+47q^{-39}+36q^{-41}+q^{-43}-23q^{-45}-26q^{-47}-11q^{-49}+6q^{-51}+14q^{-53}+9q^{-55}-3q^{-59}-4q^{-61}-2q^{-63}+q^{-65}+q^{-67}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+2q^{8}+q^{6}+q^{4}-1-q^{-4}+q^{-6}+q^{-8}$
1,1	$q^{44}-2q^{42}+4q^{40}-8q^{38}+11q^{36}-12q^{34}+12q^{32}-12q^{30}+4q^{28}+2q^{26}-8q^{24}+16q^{22}-19q^{20}+22q^{18}-20q^{16}+20q^{14}-14q^{12}+10q^{10}-2q^{8}-4q^{6}+8q^{4}-14q^{2}+14-12q^{-2}+11q^{-4}-4q^{-6}+4q^{-8}+2q^{-10}-q^{-12}-2q^{-16}+q^{-20}$
2,0	$q^{42}-q^{38}-q^{36}+q^{32}-q^{30}-q^{28}+2q^{18}+3q^{16}+q^{14}+q^{12}-q^{8}-2q^{6}-q^{4}-2q^{2}+2q^{-2}+3q^{-4}+2q^{-6}+2q^{-10}-2q^{-14}-q^{-16}+q^{-20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-q^{32}-2q^{26}+q^{24}-q^{22}-q^{20}+q^{18}+q^{16}+2q^{12}+2q^{10}+q^{8}+q^{6}+q^{2}-1-q^{-2}+q^{-4}-2q^{-6}+2q^{-10}+q^{-16}$
1,0,0	$-q^{21}-q^{17}+2q^{11}+2q^{9}+2q^{7}+q^{5}-q-2q^{-1}-q^{-5}+q^{-7}+q^{-9}+q^{-11}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}+q^{38}-q^{36}-3q^{34}-2q^{32}-q^{30}-3q^{28}-q^{26}+3q^{24}+5q^{22}+2q^{20}+3q^{18}+4q^{16}-q^{14}-2q^{12}-q^{8}-q^{6}+2q^{4}+3q^{2}+1+q^{-4}-3q^{-8}-q^{-10}+q^{-12}+q^{-18}+q^{-20}+q^{-22}$
1,0,0,0	$-q^{26}-q^{22}-q^{20}+2q^{14}+2q^{12}+3q^{10}+2q^{8}+q^{6}-q^{2}-2-2q^{-2}-q^{-6}+q^{-8}+q^{-10}+q^{-12}+q^{-14}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+q^{32}-2q^{30}+2q^{28}-2q^{26}+q^{24}-q^{22}+q^{20}+q^{18}-q^{16}+4q^{14}-2q^{12}+4q^{10}-3q^{8}+3q^{6}-2q^{4}+q^{2}-1-q^{-2}+q^{-4}-2q^{-6}+2q^{-8}-2q^{-10}+2q^{-12}+q^{-16}$
1,0	$q^{56}-q^{52}-q^{50}+q^{48}+q^{46}-2q^{44}-2q^{42}+q^{40}+2q^{38}-2q^{34}-q^{32}+2q^{30}+q^{28}-q^{24}+q^{22}+q^{20}+q^{18}-q^{16}+2q^{12}+q^{10}-q^{8}-q^{6}+q^{4}+2q^{2}-2q^{-2}+2q^{-6}-2q^{-10}-q^{-12}+q^{-14}+2q^{-16}-q^{-20}+q^{-26}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+q^{42}-2q^{40}+2q^{38}-2q^{36}-2q^{32}-q^{28}-q^{26}+q^{24}-q^{22}+3q^{20}+5q^{16}+5q^{12}-q^{10}+3q^{8}-q^{6}+q^{4}-2q^{2}-1-q^{-2}-2q^{-4}+q^{-6}-2q^{-8}+q^{-10}-q^{-12}+3q^{-14}+q^{-18}+q^{-22}$

G2 Invariants.

Weight	Invariant
1,0	$q^{80}-q^{78}+2q^{76}-3q^{74}+q^{72}-4q^{68}+6q^{66}-5q^{64}+3q^{62}-q^{60}-4q^{58}+5q^{56}-4q^{54}+2q^{50}-4q^{48}+4q^{46}-4q^{42}+7q^{40}-5q^{38}+3q^{36}-2q^{32}+6q^{30}-4q^{28}+6q^{26}-2q^{24}+2q^{22}+4q^{20}-4q^{18}+4q^{16}-2q^{14}+q^{12}+3q^{10}-4q^{8}+2q^{6}-4q^{2}+5-6q^{-2}+2q^{-6}-6q^{-8}+5q^{-10}-3q^{-12}+q^{-14}+q^{-16}-3q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-32}+q^{-38}$

.

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["9 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}-4t+7-4t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+1$

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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}-2q+3-3q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+3a^{2}-2z^{2}-2+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-10a^{2}z^{4}-3z^{4}-3a^{5}z^{3}-a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+z^{2}a^{-2}+6z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$

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 $0$ $-8$ $0$ $0$ $-8$ $0$ ${\frac {16}{3}}$ ${\frac {64}{3}}$ $-8$ $0$ $32$ $0$ $0$ $48$ $-{\frac {56}{3}}$ ${\frac {104}{3}}$ $-{\frac {16}{3}}$ $0$

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

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 χ 5               1 1 3             1   -1 1           2 1   1 -1         2 2     0 -3       1 1       0 -5     1 2         1 -7   1 1           0 -9   1             1 -11 1               -1

Integral Khovanov Homology (db, data source)

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

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[9, 44]]
Out[2]=	9
In[3]:=	PD[Knot[9, 44]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[14, 8, 15, 7], X[18, 15, 1, 16], X[16, 11, 17, 12], X[12, 17, 13, 18], X[6, 14, 7, 13]]
In[4]:=	GaussCode[Knot[9, 44]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, -9, 5, 3, -4, 2, 7, -8, 9, -5, 6, -7, 8, -6]
In[5]:=	BR[Knot[9, 44]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 44]][t]
Out[6]=	-2 4 2 7 + t - - - 4 t + t t
In[7]:=	Conway[Knot[9, 44]][z]
Out[7]=	4 1 + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 44]}
In[9]:=	{KnotDet[Knot[9, 44]], KnotSignature[Knot[9, 44]]}
Out[9]=	{17, 0}
In[10]:=	J=Jones[Knot[9, 44]][q]
Out[10]=	-5 2 2 3 3 2 3 - q + -- - -- + -- - - - 2 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 44]}
In[12]:=	A2Invariant[Knot[9, 44]][q]
Out[12]=	-16 2 -6 -4 4 6 8 -1 - q + -- + q + q - q + q + q 8 q
In[13]:=	Kauffman[Knot[9, 44]][a, z]
Out[13]=	2 -2 2 4 z 3 5 2 z 2 2 -2 - a - 3 a - a - - - a z + a z + a z + 6 z + -- + 10 a z + a 2 a 3 4 2 2 z 3 3 3 5 3 4 2 4 5 a z + ---- + 4 a z - a z - 3 a z - 3 z - 10 a z - a 4 4 5 3 5 5 5 6 2 6 4 6 7 7 a z - 3 a z - 2 a z + a z + z + 3 a z + 2 a z + a z + 3 7 a z
In[14]:=	{Vassiliev[2][Knot[9, 44]], Vassiliev[3][Knot[9, 44]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 44]][q, t]
Out[15]=	2 1 1 1 1 1 2 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 1 2 3 5 2 ---- + --- + q t + q t + q t 3 q t q t